Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7624888 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #529 ◁ #2966 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1187 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,9,10,6)(4,12,13,7)(11,14,16,15)(18,24)(19,28)(20,23)(22,29)(25,38)(27,43)(30,46)(31,45)(33,44)(34,35)(36,41)(37,39)(40,48)(42,47);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,15)(13,16)(17,18,21,24)(19,25,31,36)(20,32,23,26)(22,27,35,37)(28,41,45,38)(29,39,34,43)(30,42,33,40)(44,47,46,48);(17,19,29,44,23,30,35,45,21,31,34,46,20,33,22,28)(18,25,39,47,26,42,37,38,24,36,43,48,32,40,27,41);(3,10)(6,9)(11,16)(14,15)(18,24)(25,36)(26,32)(27,37)(38,41)(39,43)(40,42)(47,48);(1,4,5,13)(2,7,8,12)(3,11,10,16)(6,14,9,15)(17,20,21,23)(18,26,24,32)(19,30,31,33)(22,34,35,29)(25,40,36,42)(27,39,37,43)(28,44,45,46)(38,47,41,48)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=gh,e^2=f,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gh,h^a=hi,d^b=df,e^b=ei,e^c=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72287098, #72320055, #72320269, … |
| maximal normal subgroup of | #72287098, #72320055, #72320269, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532271);