Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #506414 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #28259 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #3414 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #1747 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
567 conjugacy classes of subgroups; only the 136 normal subgroups are shown.
This group has 567 conjugacy classes of subgroups — too many to draw. Only its 136 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 128 points:
(1,2,7,13)(3,15,19,9)(4,22,25,74)(5,11,30,48)(6,31,33,72)(8,35,16,14)(10,41,43,24)(12,49,51,34)(17,59,61,71)(18,38,64,54)(20,67,66,90)(21,40,57,56)(23,73,78,122)(26,79,36,50)(27,80,82,45)(28,84,87,120)(29,88,58,47)(32,52,91,44)(37,62,53,97)(39,69,55,108)(42,98,101,76)(46,104,106,86)(60,113,114,112)(63,95,70,111)(65,103,116,119)(68,93,118,96)(75,123,81,121)(77,124,89,105)(83,94,126,110)(85,107,128,100)(92,127,109,115)(99,117,102,125);(1,3,16,57,7,19,8,21)(2,9,35,56,13,15,14,40)(4,17,36,70,25,61,26,63)(5,27,29,64,30,82,58,18)(6,20,49,108,33,66,34,69)(10,37,52,96,43,53,44,93)(11,45,47,54,48,80,88,38)(12,39,72,67,51,55,31,90)(22,71,79,111,74,59,50,95)(23,75,77,114,78,81,89,60)(24,62,91,118,41,97,32,68)(28,83,86,116,87,126,104,65)(42,99,100,109,101,102,107,92)(46,103,84,110,106,119,120,94)(73,121,105,112,122,123,124,113)(76,117,85,127,98,125,128,115);(1,4,7,25)(2,10,13,43)(3,17,19,61)(5,23,30,78)(6,32,33,91)(8,36,16,26)(9,37,15,53)(11,42,48,101)(12,50,51,79)(14,52,35,44)(18,60,64,114)(20,68,66,118)(21,70,57,63)(22,72,74,31)(24,49,41,34)(27,81,82,75)(28,85,87,128)(29,89,58,77)(38,92,54,109)(39,95,55,111)(40,96,56,93)(45,102,80,99)(46,105,106,124)(47,107,88,100)(59,90,71,67)(62,108,97,69)(65,117,116,125)(73,120,122,84)(76,104,98,86)(83,115,126,127)(94,121,110,123)(103,113,119,112);(1,5)(2,11)(3,18,19,64)(4,23)(6,28)(7,30)(8,29)(9,38,15,54)(10,42)(12,46)(13,48)(14,47)(16,58)(17,60,61,114)(20,65,66,116)(21,27,57,82)(22,73)(24,76)(25,78)(26,77)(31,84)(32,85)(33,87)(34,86)(35,88)(36,89)(37,92,53,109)(39,94,55,110)(40,45,56,80)(41,98)(43,101)(44,100)(49,104)(50,105)(51,106)(52,107)(59,112,71,113)(62,115,97,127)(63,75,70,81)(67,119,90,103)(68,117,118,125)(69,83,108,126)(72,120)(74,122)(79,124)(91,128)(93,99,96,102)(95,121,111,123)Polycyclic presentation:
2^9|a^2=i,b^2=egh,c^2=i,d^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=cg,d^b=dh,e^b=ei,g^b=gi,f^c=fh,h^c=hi,g^d=gi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 413612);