Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #190028 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#381 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2406 conjugacy classes of subgroups; only the 278 normal subgroups are shown.
This group has 2406 conjugacy classes of subgroups — too many to draw. Only its 278 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 — 4 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(11,17)(12,20)(14,18)(16,22);(2,5)(6,8)(9,10)(11,18,17,14)(12,16)(13,15)(19,24,23,21)(20,22);(1,3)(2,6)(4,7)(5,8)(9,11,12,19,13,17,20,23)(10,14,16,21,15,18,22,24);(3,7)(6,8)Polycyclic presentation:
2^9|b^2=f,c^2=h,h^2=i,b^a=be,c^a=cf,h^a=hi,c^b=cg,g^b=gi,d^c=de,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72338464, #72338465, #72338501, … |
| maximal normal subgroup of | #72338464, #72338465, #72338501, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 289905);