Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509113 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #30400 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2929 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
649 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2)(3,14)(4,19)(5,9)(6,29)(7,12)(8,15)(10,21)(11,25)(13,27)(16,46)(17,31)(18,34)(20,55)(22,58)(23,54)(24,43)(26,42)(28,35)(30,41)(32,47)(33,50)(36,49)(37,56)(38,57)(39,60)(40,52)(44,45)(48,64)(51,63)(53,62)(59,61);(1,3,6,18,7,15,27,45)(2,8,13,34,12,14,29,44)(4,20,22,53,23,16,30,48)(5,24,25,50,28,17,49,52)(9,31,36,33,35,43,11,40)(10,37,39,63,38,32,42,61)(19,46,41,62,54,55,58,64)(21,47,26,51,57,56,60,59);(2,9)(3,15)(5,25)(6,27)(8,31)(10,19)(11,13)(12,35)(14,43)(16,20)(17,50)(21,26)(22,30)(24,52)(28,49)(29,36)(32,55)(33,44)(34,40)(37,46)(38,54)(39,58)(41,42)(47,59)(51,56)(57,60)(61,62)(63,64);(1,4,7,23)(2,10,12,38)(3,16,15,20)(5,26,28,60)(6,30,27,22)(8,32,14,37)(9,19,35,54)(11,41,36,58)(13,42,29,39)(17,51,24,59)(18,53,45,48)(21,49,57,25)(31,55,43,46)(33,62,40,64)(34,63,44,61)(47,50,56,52)Polycyclic presentation:
2^9|b^2=h,d^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fh,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,f^c=fh,h^c=hi,f^d=fh,h^d=hi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72320520, #72320582, #72320752 |
| maximal normal subgroup of | #72320520, #72320582, #72320752 |
| maximal quotient of | #72321394, #72321571, #72321723, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 416310);