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 | |
1598 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)(17,18,22,26)(19,28,30,23)(20,33,36,42)(21,25,37,27)(24,40,43,35)(29,45,34,48)(31,44,32,39)(38,47,41,46);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,34)(21,32)(22,30)(24,41)(25,44)(26,28)(27,39)(29,36)(31,37)(33,45)(35,46)(38,43)(40,47)(42,48);(1,3,5,4)(2,7,8,6)(10,13)(14,16)(17,20,21,35,22,36,37,40)(18,24,27,33,26,43,25,42)(19,29,32,47,30,34,31,46)(23,38,39,48,28,41,44,45);(1,3,5,4)(2,7,8,6)(9,11)(10,14)(12,15)(13,16)(19,30)(23,28)(29,34)(31,32)(38,41)(39,44)(45,48)(46,47);(1,4,5,3)(2,6,8,7)(11,15)(14,16)(17,21,22,37)(18,25,26,27)(19,31,30,32)(20,35,36,40)(23,39,28,44)(24,42,43,33)(29,46,34,47)(38,48,41,45)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=fgi,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, 6467777);