Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #14715 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C23×C4 | C24 |
| commutator, socle | C24 | C23×C4 |
| prime radical | C23×#41 | C4 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4014 conjugacy classes of subgroups; only the 482 normal subgroups are shown.
This group has 4014 conjugacy classes of subgroups — too many to draw. Only its 482 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) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(5,6)(7,8)(17,18,19,22,20,21,23,24);(2,4)(9,10)(11,14)(12,13)(15,16);(6,8)(10,13)(14,16);(9,11,12,15)(10,14,13,16)(18,21)(22,24)Polycyclic presentation:
2^9|a^2=h,d^2=g,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg128 conjugacy classes — view the full character table.
| maximal subgroup of | #72231572, #72239413, #72239700, … |
| maximal normal subgroup of | #72231572, #72239413, #72239700, … |
| maximal quotient of | #72231501, #72281379, #72282116, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266035);