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 | C23×D4 | D4 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#123 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
696 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 696 conjugacy classes of subgroups — too many to draw. Only its 72 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)(3,14,17,19)(4,16,21,32)(6,9,23,28)(7,13,25,30)(8,26,20,12)(10,15,29,31)(18,24,27,22);(1,3,15,11,6,16,24,28)(2,8,27,22,9,25,29,31)(4,12,5,17,19,13,23,32)(7,10,14,30,20,18,21,26);(2,9)(4,19)(5,23)(8,25)(12,13)(17,32)(22,31)(27,29)Polycyclic presentation:
2^9|a^2=f,b^2=dfgi,d^2=ghi,b^a=bd,c^a=ce,d^a=dg,e^a=egh,g^a=gi,h^a=hi,d^b=dh,e^b=egh,f^b=fhi,g^b=gi,h^b=hi,d^c=dgh,f^c=fgh,f^d=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72238460, #72238463, #72238468, … |
| maximal normal subgroup of | #72238460, #72238463, #72238468, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 46654);