Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #149753 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #11500 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | #381 | C23 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C25 ◁ #2308 ◁ G | |
| LECS | G ▷ #381 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
483 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 483 conjugacy classes of subgroups — too many to draw. Only its 60 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:
(1,2,5,9)(3,11,13,28,14,27,30,32)(4,15,17,25)(6,10,18,26)(7,19,20,29,21,31,24,12)(8,22,23,16);(1,3)(2,7,10,21)(4,12)(5,13)(6,14)(8,11,15,27)(9,24,26,20)(16,29)(17,19)(18,30)(22,31)(23,32,25,28);(4,16)(8,15)(11,27)(12,29)(17,22)(19,31)(23,25)(28,32)Polycyclic presentation:
2^9|a^2=e,b^2=h,d^2=i,e^2=g,b^a=bd,c^a=cg,d^a=df,f^a=fh,h^a=hi,e^b=efi,g^b=gi,d^c=di,e^d=eh,f^e=fi41 conjugacy classes — view the full character table.
| maximal subgroup of | #72237387, #72237722, #72237726, … |
| maximal normal subgroup of | #72237387, #72237722, #72237726, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56949);