Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907993 |
| 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 |
| prime radical | #30708 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1468 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,7)(3,8,4,6)(17,18,22,26)(19,27,30,23)(20,31,34,38)(21,35,36,25)(24,32,40,28)(29,37,43,41)(33,46,47,44)(39,42,48,45);(1,2)(3,6)(4,8)(5,7)(9,10)(11,15)(12,14)(13,16)(17,19,22,30)(18,23,26,27)(20,32,34,28)(21,29,36,43)(24,38,40,31)(25,37,35,41)(33,45,47,42)(39,46,48,44);(1,2)(3,8)(4,6)(5,7)(9,11)(10,13)(12,16)(14,15)(17,20,19,28,22,34,30,32)(18,24,23,31,26,40,27,38)(21,33,29,42,36,47,43,45)(25,39,37,44,35,48,41,46);(1,3,5,4)(2,6,7,8)(11,16)(13,15)(17,21,22,36)(18,25,26,35)(19,29,30,43)(20,33,34,47)(23,37,27,41)(24,39,40,48)(28,42,32,45)(31,44,38,46);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(21,36)(25,35)(29,43)(33,47)(37,41)(39,48)(42,45)(44,46)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fgi,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=dfh,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6822642);