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 | #36011 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1492 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)(11,15)(12,16)(13,14)(17,18)(19,24)(20,26)(21,34)(22,25)(23,27)(28,42)(29,37)(30,38)(31,36)(32,41)(33,39)(35,40)(43,48)(44,47)(45,46);(1,3,4,8)(2,5,6,7)(17,19,22,30)(18,24,25,38)(20,29,33,44)(21,31,23,28)(26,37,39,47)(27,42,34,36)(32,43,35,45)(40,46,41,48);(10,14)(15,16)(18,25)(19,28)(21,23)(24,36)(26,39)(29,43)(30,31)(32,35)(37,46)(38,42)(44,45)(47,48);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(17,20)(18,26)(19,29)(21,35)(22,33)(23,32)(24,37)(25,39)(27,41)(28,43)(30,44)(31,45)(34,40)(36,46)(38,47)(42,48);(3,8)(5,7)(9,12,13,11)(10,16,14,15)(17,21)(18,27)(20,32)(22,23)(24,38)(25,34)(26,40)(28,31)(29,44)(33,35)(39,41)(46,48)Polycyclic presentation:
2^9|b^2=fi,d^2=gi,e^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,e^c=ei,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, 6637387);