Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7234345 |
| 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 | #42908 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1781 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)(3,7)(4,6)(5,8)(17,18,24,30)(19,25,33,42)(20,27,21,36)(22,29,23,28)(26,37,44,35)(31,46,34,40)(32,45,38,41)(39,47,43,48);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(17,19)(18,25)(20,34)(21,31)(22,38)(23,32)(24,33)(26,43)(27,40)(28,45)(29,41)(30,42)(35,47)(36,46)(37,48)(39,44);(9,11)(10,13)(12,16)(14,15)(17,20)(18,26)(19,31)(21,24)(22,35)(23,37)(25,39)(27,28)(29,36)(30,44)(32,47)(33,34)(38,48)(40,41)(42,43)(45,46);(3,8)(5,7)(11,16)(13,15)(17,21,24,20)(18,27,30,36)(19,31,33,34)(22,35,23,37)(25,40,42,46)(26,29,44,28)(32,48,38,47)(39,45,43,41);(2,6)(5,7)(10,14)(13,15)(17,22,24,23)(18,28,30,29)(19,32,33,38)(20,35,21,37)(25,41,42,45)(26,27,44,36)(31,47,34,48)(39,40,43,46)Polycyclic presentation:
2^9|a^2=i,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=efi,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7141612);