Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #513025 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #33669 |
| commutator, Frattini, cosocle | C2×C16 | C24 |
| prime radical | #3414 | C22 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ #2108 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
336 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 336 conjugacy classes of subgroups — too many to draw. Only its 106 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:
(1,2,8,14)(3,9,16,39)(4,20,22,74)(5,26,29,83)(6,33,36,96)(7,37,35,13)(10,43,45,23)(11,48,51,31)(12,53,55,30)(15,56,58,72)(17,63,27,82)(18,68,61,109)(19,70,62,42)(21,73,25,81)(24,80,77,112)(28,54,91,105)(32,94,92,85)(34,50,76,111)(38,78,98,59)(40,88,49,66)(41,87,100,65)(44,75,47,79)(46,95,106,90)(52,93,84,107)(57,113,71,114)(60,118,116,126)(64,103,86,125)(67,124,89,120)(69,101,119,108)(97,115,104,117)(99,121,128,122)(102,123,110,127);(1,3,8,16)(2,9,14,39)(4,15,22,58)(5,27,29,17)(6,18,36,61)(7,19,35,62)(10,38,45,98)(11,49,51,40)(12,41,55,100)(13,42,37,70)(20,72,74,56)(21,71,25,57)(23,78,43,59)(24,60,77,116)(26,82,83,63)(28,86,91,64)(30,65,53,87)(31,88,48,66)(32,89,92,67)(33,68,96,109)(34,69,76,119)(44,104,47,97)(46,99,106,128)(50,108,111,101)(52,110,84,102)(54,103,105,125)(73,113,81,114)(75,115,79,117)(80,126,112,118)(85,124,94,120)(90,121,95,122)(93,127,107,123);(1,4)(2,10)(3,15)(5,28)(6,34)(7,21)(8,22)(9,38)(11,50)(12,54)(13,44)(14,45)(16,58)(17,64)(18,69)(19,71)(20,26)(23,31)(24,30)(25,35)(27,86)(29,91)(32,95)(33,46)(36,76)(37,47)(39,98)(40,101)(41,103)(42,104)(43,48)(49,108)(51,111)(52,112)(53,77)(55,105)(56,63)(57,62)(59,66)(60,65)(61,119)(67,121)(68,99)(70,97)(72,82)(73,94)(74,83)(75,93)(78,88)(79,107)(80,84)(81,85)(87,116)(89,122)(90,92)(96,106)(100,125)(102,126)(109,128)(110,118)(113,120)(114,124)(115,127)(117,123);(3,16)(4,21,22,25)(5,29)(9,39)(10,44,45,47)(11,51)(15,57,58,71)(18,61)(19,62)(20,73,74,81)(23,79,43,75)(24,76,77,34)(26,83)(28,90,91,95)(31,48)(32,92)(38,97,98,104)(41,100)(42,70)(46,105,106,54)(50,80,111,112)(52,84)(56,113,72,114)(59,117,78,115)(60,69,116,119)(64,121,86,122)(65,87)(68,109)(85,94)(93,107)(99,103,128,125)(101,126,108,118)Polycyclic presentation:
2^9|a^2=i,b^2=i,d^2=e,f^2=gh,g^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fg,g^a=gh,h^a=hi,d^b=di,f^b=fi,d^c=dh,e^c=ei,f^c=fg,g^c=gh,h^c=hi,f^d=fi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420223);