Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667580 |
| 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 | #23871 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1627 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)(9,10)(11,14)(12,13)(15,16)(17,18)(19,28)(20,33)(21,26)(22,27)(23,25)(24,32)(29,40)(30,38)(31,45)(34,44)(35,48)(36,43)(37,41)(39,47)(42,46);(1,3)(2,5)(4,8)(6,7)(17,19)(18,24)(20,30)(21,35)(22,32)(23,29)(25,38)(26,42)(27,28)(31,37)(33,40)(34,47)(36,46)(39,41)(43,48)(44,45);(10,13)(14,16)(19,29)(20,23)(21,36)(24,38)(25,33)(26,43)(28,40)(30,32)(31,35)(39,42)(45,48)(46,47);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,20,22,23)(18,25,27,33)(19,30,32,29)(21,34,36,37)(24,38,28,40)(26,41,43,44)(31,35,47,46)(39,48,45,42);(3,8)(5,7)(17,21,22,36)(18,26,27,43)(19,31,32,47)(20,34,23,37)(24,39,28,45)(25,41,33,44)(29,46,30,35)(38,48,40,42)Polycyclic presentation:
2^9|a^2=fi,d^2=gi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6602803);