Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #133885 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1781 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C23×D4 | C23 |
| prime radical | #35528 | C2 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#127 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
1221 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 1221 conjugacy classes of subgroups — too many to draw. Only its 86 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)(11,17,19,23)(12,20,21,15)(13,16)(14,18)(22,24);(9,11)(10,14,16,22)(12,18)(13,19)(15,17,20,23)(21,24);(1,3)(2,5)(4,8)(6,7)(12,21)(15,20)(17,23)(18,24)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,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 | #72233270, #72236834, #72261255, … |
| maximal normal subgroup of | #72233270, #72236834, #72261255, … |
| maximal quotient of | #72233266, #72238355, #72238356 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 41081);