Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #213744 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2308 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
1686 conjugacy classes of subgroups; only the 234 normal subgroups are shown.
This group has 1686 conjugacy classes of subgroups — too many to draw. Only its 234 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:
(1,2,4,7)(3,6,8,5)(9,10)(11,14)(12,15)(13,16)(17,22)(18,21)(19,23)(20,24);(1,3)(2,5)(4,8)(6,7)(9,11)(10,14)(12,19,20,17)(13,18)(15,23,24,22)(16,21);(1,3)(2,6)(4,8)(5,7)(14,21)(22,23);(3,8)(5,6)(9,12)(10,15)(11,17)(13,20)(14,22)(16,24)(18,19)(21,23)Polycyclic presentation:
2^9|a^2=e,b^2=h,b^a=be,c^a=cf,g^a=gi,c^b=cg,d^b=def,f^b=fi,d^c=de,h^d=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72309322, #72309335, #72309384, … |
| maximal normal subgroup of | #72309322, #72309335, #72309384, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 298931);