Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| 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 | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1583 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)(9,10,13,15)(11,14,12,16)(17,18)(19,27)(20,25)(21,24)(22,26)(23,30)(28,37)(29,45)(31,43)(32,48)(33,39)(34,42)(35,40)(36,44)(38,47)(41,46);(1,3,4,8)(2,5,6,7)(9,11,13,12)(10,14,15,16)(17,19)(18,23)(20,32)(21,30)(22,28)(24,27)(25,41)(26,43)(29,36)(31,35)(33,46)(34,47)(37,40)(38,44)(39,48)(42,45);(9,12,13,11)(10,14,15,16)(18,24)(19,28)(22,35)(23,37)(25,39)(27,43)(29,46)(30,31)(32,47)(34,36)(38,41)(45,48);(2,6)(5,7)(20,33)(25,39)(29,47)(32,46)(34,36)(38,45)(41,48)(42,44);(3,8)(5,7)(17,20,21,33)(18,25,24,39)(19,29,30,47)(22,36,35,34)(23,38,27,45)(26,44,40,42)(28,46,31,32)(37,41,43,48)Polycyclic presentation:
2^9|a^2=gi,b^2=fg,c^2=gi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6600149);