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 | #32712 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1653 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,6)(3,7,8,4)(17,18)(19,27)(20,32)(21,23)(22,35)(24,31)(25,33)(26,29)(28,40)(30,44)(34,38)(36,42)(37,48)(39,41)(43,47)(45,46);(2,6)(4,7)(9,10)(11,15)(12,14)(13,16)(18,23)(19,28)(20,33)(26,35)(31,42)(34,43)(37,46)(39,44);(1,3)(2,7)(4,6)(5,8)(9,11)(10,13)(12,16)(14,15)(17,19,29,42,21,28,22,31)(18,24,35,40,23,36,26,27)(20,30,43,48,33,41,34,45)(25,37,47,44,32,46,38,39);(2,6)(3,8)(11,16)(13,15)(17,20)(18,25)(19,30)(21,33)(22,34)(23,32)(24,37)(26,38)(27,39)(28,41)(29,43)(31,45)(35,47)(36,46)(40,44)(42,48);(1,4,5,7)(2,3,6,8)(10,14)(13,15)(19,28)(20,33)(24,36)(25,32)(27,40)(31,42)(34,43)(38,47)Polycyclic presentation:
2^9|a^2=fi,c^2=gi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=dfh,e^c=ei,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6819866);