Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #508070 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1226 conjugacy classes of subgroups; only the 124 normal subgroups are shown.
This group has 1226 conjugacy classes of subgroups — too many to draw. Only its 124 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)(3,6)(4,5)(7,8)(10,14)(12,19)(15,21)(16,20)(18,23)(22,24);(2,5)(6,8)(11,17)(12,19)(15,22)(21,24);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,20)(13,16)(14,19)(17,22)(18,24)(21,23);(3,7)(6,8)(9,11,13,17)(10,15,16,22)(12,18,19,23)(14,21,20,24)Polycyclic presentation:
2^9|d^2=i,f^2=i,b^a=be,c^a=cf,f^a=fi,g^a=ghi,c^b=cg,d^b=di,f^b=fh,d^c=de,f^c=fi,h^c=hi,g^f=gi41 conjugacy classes — view the full character table.
| maximal subgroup of | #72285728, #72285851, #72286113, … |
| maximal normal subgroup of | #72285728, #72285851, #72286113, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 415289);