Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535664 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1591 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,7,4,6)(17,18,22,27)(19,28,30,24)(20,32,35,46)(21,26,23,38)(25,34,43,36)(29,44,33,47)(31,41,37,40)(39,45,42,48);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,33)(21,37)(22,30)(23,31)(25,42)(26,41)(27,28)(29,35)(32,44)(34,45)(36,48)(38,40)(39,43)(46,47);(1,3,5,4)(2,7,8,6)(10,13)(14,16)(17,20,21,34,22,35,23,36)(18,25,38,46,27,43,26,32)(19,29,37,48,30,33,31,45)(24,39,40,44,28,42,41,47);(1,3,5,4)(2,7,8,6)(9,11)(10,14)(12,15)(13,16)(19,30)(24,28)(29,33)(31,37)(39,42)(40,41)(44,47)(45,48);(1,4,5,3)(2,6,8,7)(11,15)(14,16)(17,21,22,23)(18,26,27,38)(19,31,30,37)(20,34,35,36)(24,40,28,41)(25,32,43,46)(29,45,33,48)(39,44,42,47)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=fg,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ei,e^d=ehi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6467774);