Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #184767 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C22×#534 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C22⋊C4 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
5738 conjugacy classes of subgroups; only the 650 normal subgroups are shown.
This group has 5738 conjugacy classes of subgroups — too many to draw. Only its 650 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 — 4 generators on 24 points:
(1,2)(3,4)(9,10)(11,17,19,14)(12,20,21,15)(13,16)(18,22)(23,24);(2,4)(5,6)(7,8)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24);(6,8)(9,12)(10,15)(11,18)(13,21)(14,22)(16,20)(17,23)(19,24);(11,19)(14,17)(18,24)(22,23)Polycyclic presentation:
2^9|a^2=h,b^a=be,c^a=cf,e^a=ei,f^a=fi,c^b=cg,d^b=di,h^b=hi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72240560, #72240570, #72240598, … |
| maximal normal subgroup of | #72240560, #72240570, #72240598, … |
| maximal quotient of | #72258613, #72261492, #72261803, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264656);