Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667932 |
| 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 | #2407 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1477 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,7,8,5)(17,18,23,28)(19,24,29,38)(20,26,34,42)(21,27,22,25)(30,37,43,48)(31,36,32,39)(33,40,35,41)(44,47,45,46);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,16)(13,14)(17,19)(18,24)(20,30)(21,31)(22,32)(23,29)(25,39)(26,37)(27,36)(28,38)(33,44)(34,43)(35,45)(40,47)(41,46)(42,48);(10,14)(15,16)(18,25)(19,29)(21,22)(24,36)(26,40)(27,28)(30,43)(33,35)(37,46)(38,39)(41,42)(47,48);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(17,20)(18,26)(19,30)(21,35)(22,33)(23,34)(24,37)(25,40)(27,41)(28,42)(29,43)(31,45)(32,44)(36,46)(38,48)(39,47);(3,8)(5,7)(9,12,13,11)(10,15,14,16)(17,21,23,22)(18,27,28,25)(19,31,29,32)(20,33,34,35)(24,36,38,39)(26,40,42,41)(30,44,43,45)(37,47,48,46)Polycyclic presentation:
2^9|a^2=fi,d^2=hi,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,e^c=ei,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6766583);