Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #122191 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1446 |
| commutator | C25 | C22×C4 |
| Frattini, prime radical, cosocle | C26 | C23 |
| LCS | G ▷ C25 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C26 ◁ G | |
| LECS | G ▷ C26 ▷ C22 ▷ C1 | |
1400 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 1400 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) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(9,10)(11,17)(12,20)(13,16)(14,19)(15,21)(18,22)(23,24);(1,3)(2,5)(4,8)(6,7)(9,11)(10,14,16,17)(12,18)(13,19)(15,23,20,22)(21,24);(1,2)(3,5)(4,6)(7,8)(9,12)(10,15,16,20)(11,18)(13,21)(14,22,17,23)(19,24)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=f,b^a=bd,c^a=ce,d^a=dh,f^a=fi,c^b=cf,d^b=di,e^b=eh,g^b=gh,d^c=dhi,e^c=ei56 conjugacy classes — view the full character table.
| maximal subgroup of | #72226618, #72232805, #72233110, … |
| maximal normal subgroup of | #72226618, #72232805, #72233110, … |
| maximal quotient of | #72232803, #72233007, #72236952, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 29387);