Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139432 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1898 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×C22⋊C4 | C23 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#121 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
785 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 785 conjugacy classes of subgroups — too many to draw. Only its 74 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:
(1,2,5,11)(3,13,17,22)(4,18,21,32)(6,15,25,31)(7,12,26,30)(8,27,20,9)(10,14,29,19)(16,24,28,23);(1,3,14,11,6,18,24,31)(2,8,28,23,15,26,29,19)(4,9,5,17,22,12,25,32)(7,10,13,30,20,16,21,27);(2,9)(4,19)(5,8)(12,15)(17,28)(22,23)(25,26)(29,32)Polycyclic presentation:
2^9|a^2=f,b^2=dfgi,d^2=ghi,b^a=bd,c^a=ce,d^a=dg,e^a=eghi,g^a=gi,h^a=hi,d^b=dh,e^b=eghi,f^b=fhi,g^b=gi,h^b=hi,d^c=dghi,f^c=fghi,f^d=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72233255, #72233279, #72233283, … |
| maximal normal subgroup of | #72233255, #72233279, #72233283, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 46670);