Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #103197 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #392 |
| commutator, Frattini, cosocle | C24×C4 | C23 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C24×C4 ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C23 ▷ C1 | |
911 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 911 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 — 3 generators on 24 points:
(2,6)(3,4)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,21,24)(19,22,20,23);(1,2,5,7)(3,8,4,6)(9,11)(10,13)(12,15)(14,16)(18,22)(19,20)(23,24);(1,2)(3,6)(4,8)(5,7)(13,16)(18,23)(19,20)(22,24)Polycyclic presentation:
2^9|a^2=h,b^2=gh,d^2=g,e^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fhi,c^b=cf,d^b=dg,e^b=ei,d^c=dh,e^c=eg50 conjugacy classes — view the full character table.
| maximal subgroup of | #72264388, #72264959, #72264969 |
| maximal normal subgroup of | #72264388, #72264959, #72264969 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10417);