Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #485065 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×C22⋊C4 | C24 |
| prime radical | #534 | C23 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
776 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 776 conjugacy classes of subgroups — too many to draw. Only its 120 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 48 points:
(1,2)(3,9)(4,12)(5,7)(6,10)(8,14)(11,16)(13,15)(17,18)(19,24)(20,31)(21,23)(22,27)(25,34)(26,35)(28,42)(29,37)(30,41)(32,39)(33,43)(36,40)(38,46)(44,48)(45,47);(1,3,5,11)(2,6,7,15)(4,13,14,10)(8,16,12,9)(18,23)(21,35)(24,37)(25,38)(26,27)(29,36)(31,43)(33,46)(39,47)(40,41)(42,48)(44,45);(17,19,22,30)(18,24,27,41)(20,32,34,28)(21,36,35,29)(23,37,26,40)(25,42,31,39)(33,44,46,45)(38,48,43,47);(2,7)(6,15)(8,12)(9,16)(17,20,22,34)(18,25,27,31)(19,28,30,32)(21,33,35,46)(23,38,26,43)(24,39,41,42)(29,44,36,45)(37,47,40,48)Polycyclic presentation:
2^9|b^2=gi,c^2=i,d^2=i,e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,h^a=hi,e^b=eg,f^b=fhi,d^c=di,e^c=eh,f^e=fi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 392304);