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 | C2×#449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1643 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)(9,10,12,13)(11,14,15,16)(17,18,23,27)(19,24,32,41)(20,25,36,44)(21,26,37,22)(28,39,45,47)(29,31,40,30)(33,42,46,48)(34,35,43,38);(1,3,4,8)(2,5,6,7)(10,13)(11,15)(17,19)(18,24)(20,33)(21,29)(22,30)(23,32)(25,42)(26,31)(27,41)(28,35)(34,47)(36,46)(37,40)(38,45)(39,43)(44,48);(10,13)(14,16)(17,20,23,36)(18,25,27,44)(19,28,32,45)(21,34,37,43)(22,38,26,35)(24,39,41,47)(29,42,40,48)(30,33,31,46);(2,6)(5,7)(10,13)(14,16)(17,21)(18,26)(19,29)(20,34)(22,27)(23,37)(24,31)(25,35)(28,42)(30,41)(32,40)(33,47)(36,43)(38,44)(39,46)(45,48);(3,8)(5,7)(9,11,12,15)(10,14,13,16)(19,30)(21,37)(22,26)(24,29)(28,33)(31,32)(34,43)(35,38)(39,42)(40,41)(45,46)(47,48)Polycyclic presentation:
2^9|a^2=gi,b^2=fg,c^2=i,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=df,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, 6625463);