Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #133835 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1781 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C23×D4 | C23 |
| prime radical | #35527 | C2 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#127 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
1782 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 1782 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(9,10)(11,17,19,24)(12,20,22,15)(13,16)(14,18)(21,23);(9,11)(10,14)(12,21)(13,19)(15,24)(16,23)(17,20)(18,22);(1,3)(2,5)(4,8)(6,7)(12,22)(15,20)(17,24)(18,21)Polycyclic presentation:
2^9|a^2=f,d^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,d^b=di,e^b=ei,f^b=fgi,d^c=di,f^c=fh,f^d=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72210498, #72210529, #72210539, … |
| maximal normal subgroup of | #72210498, #72210529, #72210539, … |
| maximal quotient of | #72238354, #72272668, #72310562, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 41031);