Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8516 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #379 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C22×C42 | C23 |
| prime radical | C24×D4 | C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C23 ▷ C1 | |
2080 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 2080 conjugacy classes of subgroups — too many to draw. Only its 142 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) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,7,8,5)(4,6)(9,10,11,13,12,14,15,16)(17,18)(19,26,28,23)(20,22)(21,25)(24,29)(27,31,32,30);(1,3)(2,5)(4,8)(6,7)(18,22)(20,29)(23,30)(24,25)(26,31)(27,32);(17,19)(18,23)(20,27)(21,28)(22,30)(24,31)(25,26)(29,32)Polycyclic presentation:
2^9|a^2=f,d^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,d^b=dh,f^b=fgh,f^c=fh68 conjugacy classes — view the full character table.
| maximal subgroup of | #72292811, #72338268, #72338269, … |
| maximal normal subgroup of | #72292811, #72338268, #72338269, … |
| maximal quotient of | #72292831, #72293130, #72293159, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4212);