Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499849 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
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 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1883 conjugacy classes of subgroups; only the 132 normal subgroups are shown.
This group has 1883 conjugacy classes of subgroups — too many to draw. Only its 132 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)(3,7)(4,6)(5,8)(9,10)(11,14)(12,19)(13,15)(16,20)(17,22)(18,23)(21,24);(1,3,4,8)(2,5,6,7)(11,17)(12,20)(14,22)(16,19)(18,21)(23,24);(9,11)(10,14)(12,21)(13,18)(15,23)(16,24)(17,20)(19,22);(2,6)(5,7)(10,15)(14,23)(16,19)(22,24)Polycyclic presentation:
2^9|b^2=e,g^2=i,b^a=be,c^a=cf,d^a=dei,g^a=ghi,c^b=cg,f^b=fh,g^b=gi,g^c=gi,h^c=hi,g^f=gi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72233169, #72233180, #72233427, … |
| maximal normal subgroup of | #72233169, #72233180, #72233427, … |
| maximal quotient of | #72233149, #72233174, #72233404, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407211);