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 | #2619 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1474 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)(11,15)(12,16)(13,14)(17,18,22,23)(19,27,30,24)(20,31,34,25)(21,35,36,26)(28,37,42,32)(29,38,43,40)(33,39,46,44)(41,45,48,47);(1,3)(2,5)(4,8)(6,7)(17,19,22,30)(18,24,23,27)(20,32,34,37)(21,29,36,43)(25,28,31,42)(26,38,35,40)(33,45,46,47)(39,41,44,48);(10,14)(15,16)(17,20,22,34)(18,25,23,31)(19,28,30,42)(21,33,36,46)(24,37,27,32)(26,39,35,44)(29,41,43,48)(38,47,40,45);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(17,21)(18,26)(19,29)(20,33)(22,36)(23,35)(24,38)(25,39)(27,40)(28,41)(30,43)(31,44)(32,45)(34,46)(37,47)(42,48);(3,8)(5,7)(9,12,13,11)(10,16,14,15)(18,23)(19,24)(20,34)(21,36)(27,30)(28,32)(29,40)(37,42)(38,43)(39,44)(41,47)(45,48)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,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, 6636879);