Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #93520 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #342 |
| commutator | C23×C4 | C42 |
| Frattini, cosocle | C22×C4×D4 | C22 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4×D4 ▷ C24 ▷ C2 ▷ C1 | |
782 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 782 conjugacy classes of subgroups — too many to draw. Only its 84 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) = 128 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,2)(3,5)(4,7)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18,19,22)(20,23,24,21);(2,5,7,6)(9,11)(10,13,14,15)(12,16)(17,18)(19,22,24,21)(20,23)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fh,b^a=bc,c^a=cf,e^a=efgh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=dh,f^b=fi,g^b=gi,h^b=hi,d^c=di,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72227977, #72227982 |
| maximal normal subgroup of | #72227977, #72227982 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 716);