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 | #1999 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1470 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)(17,18,23,22)(19,24,30,29)(20,25,33,36)(21,26,34,35)(27,37,41,31)(28,38,42,43)(32,39,45,46)(40,47,48,44);(1,3)(2,5)(4,8)(6,7)(17,19,23,30)(18,24,22,29)(20,31,33,37)(21,28,34,42)(25,27,36,41)(26,38,35,43)(32,44,45,47)(39,40,46,48);(10,14)(15,16)(17,20,23,33)(18,25,22,36)(19,27,30,41)(21,32,34,45)(24,37,29,31)(26,39,35,46)(28,40,42,48)(38,47,43,44);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(17,21)(18,26)(19,28)(20,32)(22,35)(23,34)(24,38)(25,39)(27,40)(29,43)(30,42)(31,44)(33,45)(36,46)(37,47)(41,48);(3,8)(5,7)(9,12,13,11)(10,16,14,15)(18,22)(19,29)(20,33)(21,34)(24,30)(27,37)(28,38)(31,41)(39,46)(40,44)(42,43)(47,48)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=i,d^2=g,e^2=g,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, 6636892);