Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153319 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13448 |
| commutator, Frattini, cosocle | C22×C16 | C23 |
| prime radical | C2×#508 | C22 |
| LCS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ #2108 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
250 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 250 conjugacy classes of subgroups — too many to draw. Only its 66 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 — 3 generators on 128 points:
(1,2)(3,19)(4,27)(5,35)(6,39)(7,13)(8,46)(9,49)(10,18)(11,21)(12,30)(14,45)(15,38)(16,36)(17,48)(20,76)(22,54)(23,72)(24,52)(25,66)(26,71)(28,95)(29,99)(31,90)(32,107)(33,59)(34,67)(37,70)(40,55)(41,62)(42,69)(43,65)(44,64)(47,63)(50,75)(51,108)(53,109)(56,84)(57,119)(58,105)(60,106)(61,77)(68,89)(73,83)(74,79)(78,111)(80,112)(81,114)(82,113)(85,125)(86,118)(87,117)(88,116)(91,102)(92,103)(93,100)(94,101)(96,120)(97,122)(98,121)(104,124)(110,126)(115,127)(123,128);(1,3,10,26)(2,11,18,56)(4,28,34,98)(5,23,38,87)(6,40,44,83)(7,21,43,84)(8,47,42,109)(9,50,48,25)(12,57,62,122)(13,63,65,53)(14,66,70,75)(15,71,35,19)(16,72,68,117)(17,73,49,55)(20,77,82,31)(22,37,88,45)(24,89,86,36)(27,76,67,113)(29,80,103,128)(30,81,41,108)(32,78,104,127)(33,79,101,85)(39,52,64,118)(46,54,69,116)(51,94,114,59)(58,123,93,112)(60,125,91,74)(61,115,90,111)(92,121,99,95)(96,106,110,102)(97,105,119,100)(107,126,124,120);(1,4,5,29,36,100,6,41,9,33,37,102,42,104,43,77,10,34,38,103,89,105,44,30,48,101,45,106,8,32,7,31)(2,12,13,58,46,92,14,67,17,61,64,124,68,91,15,59,18,62,65,93,69,99,70,27,49,90,39,107,16,60,35,94)(3,20,21,78,47,110,22,85,25,81,83,119,86,128,87,98,26,82,84,127,109,96,88,79,50,108,40,97,24,80,23,28)(11,51,19,74,72,120,52,115,55,113,75,121,116,123,53,57,56,114,71,125,117,126,118,111,73,76,66,95,54,112,63,122)Polycyclic presentation:
2^9|b^2=i,c^2=d,d^2=gh,e^2=gi,f^2=g,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eghi,f^a=fghi,g^a=gh,h^a=hi,c^b=cf,d^b=dg,e^b=eghi,f^b=fgh,g^b=gh,h^b=hi,e^c=ei56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60516);