Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #463891 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16495 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
795 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)(3,14,19,41)(4,21,25,9)(5,26)(6,11)(7,13)(8,18,33,20)(10,29)(12,15)(16,34)(17,48,51,43)(22,31)(23,35,55,53)(24,37,30,36)(27,38,58,57)(28,40)(32,52,56,50)(39,44)(42,47)(45,63)(46,60)(49,59)(54,62)(61,64);(1,3,15,20,7,19,6,18)(2,8,11,14,13,33,12,41)(4,22,24,16,25,46,30,47)(5,17,44,52,29,51,28,50)(9,34,37,31,21,42,36,60)(10,32,39,43,26,56,40,48)(23,54,27,45,55,49,58,64)(35,61,57,59,53,63,38,62);(1,4,15,24,7,25,6,30)(2,9,11,37,13,21,12,36)(3,16,20,46,19,47,18,22)(5,27,44,55,29,58,28,23)(8,31,14,42,33,60,41,34)(10,38,39,35,26,57,40,53)(17,49,52,64,51,54,50,45)(32,61,43,59,56,63,48,62);(1,5,7,29)(2,10)(3,17,19,51)(4,23)(6,28,15,44)(8,32)(9,35,21,53)(11,39)(12,40)(13,26)(14,43)(16,45)(18,50,20,52)(22,54)(24,27)(25,55)(30,58)(31,59,60,62)(33,56)(34,61,42,63)(36,38,37,57)(41,48)(46,49)(47,64)Polycyclic presentation:
2^9|a^2=efg,b^2=ef,c^2=ef,d^2=f,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,g^b=gi,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 371087);