Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150615 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #11808 |
| commutator | C2×C22⋊C4 | C22×C4 |
| Frattini, cosocle | #381 | C23 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C2×C22⋊C4 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C25 ◁ #2308 ◁ G | |
| LECS | G ▷ #381 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
467 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 467 conjugacy classes of subgroups — too many to draw. Only its 54 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,10,6,8,19,23)(3,11,13,27,14,28,30,32)(4,15,18,31,17,9,24,25)(7,20,22,12,21,29,26,16);(1,3)(2,7)(4,16,17,12)(5,13)(6,14)(8,21)(9,11,15,28)(10,26)(18,29,24,20)(19,30)(22,23)(25,27,31,32);(2,8)(4,17)(5,19)(7,21)(12,16)(13,30)(25,31)(27,32)Polycyclic presentation:
2^9|a^2=e,b^2=f,d^2=gh,e^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,g^a=gi,d^b=dg,e^b=efgh,g^b=gi,h^b=hi,d^c=di,e^c=ei,e^d=eh,f^d=fi35 conjugacy classes — view the full character table.
| maximal subgroup of | #72237478, #72237824, #72237825, … |
| maximal normal subgroup of | #72237478, #72237824, #72237825, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57813);