Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11799 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C2×#97 | C23 |
| socle | C22 | C2×#323 |
| prime radical | C24 | C22⋊C8 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
372 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 372 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 128 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2)(3,9,11,16)(4,8)(5,7)(6,10,14,12)(13,15)(17,18,20,25,21,26,32,40,22,24,31,37,33,38,44,47)(19,27,29,39,30,41,43,48,28,23,34,35,42,36,45,46);(1,3,5,11)(2,6,7,14)(4,12,13,10)(8,16,15,9)(17,19)(18,23)(20,29)(21,30)(22,28)(24,27)(25,39)(26,36)(31,34)(32,43)(33,42)(35,37)(38,41)(40,48)(44,45)(46,47);(2,7)(6,14)(8,15)(9,16)(19,28)(23,27)(29,34)(30,42)(35,39)(36,41)(43,45)(46,48)Polycyclic presentation:
2^9|a^2=e,b^2=g,d^2=g,e^2=h,h^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fi,c^b=ci,d^b=dg,e^b=efg,e^d=ei56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31752);