Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667900 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1776 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)(17,18,22,27)(19,29,32,24)(20,26,21,35)(23,38,37,28)(25,36,42,34)(30,44,31,40)(33,41,48,45)(39,46,43,47);(1,3,4,8)(2,5,6,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,30)(21,31)(22,32)(23,33)(25,39)(26,40)(27,29)(28,41)(34,46)(35,44)(36,47)(37,48)(38,45)(42,43);(10,13)(14,16)(17,20,22,21)(18,25,27,42)(19,30,32,31)(23,36,37,34)(24,39,29,43)(26,28,35,38)(33,47,48,46)(40,41,44,45);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,21,22,20)(18,26,27,35)(19,31,32,30)(23,36,37,34)(24,40,29,44)(25,28,42,38)(33,47,48,46)(39,41,43,45);(3,8)(5,7)(10,13)(14,16)(19,32)(20,21)(24,29)(25,42)(26,35)(33,48)(34,36)(41,45)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=i,d^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,e^c=ei,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6674814);