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 | #39382 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1464 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)(19,27)(20,31)(21,23)(22,35)(24,30)(25,32)(26,34)(28,40)(29,43)(33,38)(36,42)(37,48)(39,41)(44,45)(46,47);(1,2)(3,6)(4,8)(5,7)(9,10)(11,15)(12,14)(13,16)(18,23)(19,28)(20,32)(26,35)(30,42)(33,47)(37,45)(39,43);(1,2)(3,8)(4,6)(5,7)(9,11)(10,13)(12,16)(14,15)(17,19,22,30,21,28,34,42)(18,24,26,27,23,36,35,40)(20,29,33,44,32,41,47,48)(25,37,38,39,31,45,46,43);(1,3,5,4)(2,6,7,8)(11,16)(13,15)(17,20)(18,25)(19,29)(21,32)(22,33)(23,31)(24,37)(26,38)(27,39)(28,41)(30,44)(34,47)(35,46)(36,45)(40,43)(42,48);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(20,32)(25,31)(29,41)(33,47)(37,45)(38,46)(39,43)(44,48)Polycyclic presentation:
2^9|a^2=fi,c^2=fgi,d^2=fi,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^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6822716);