Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13357 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×Q8 | C24 |
| socle | C22 | C2×B(4,𝔽2) |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
1069 conjugacy classes of subgroups; only the 134 normal subgroups are shown.
This group has 1069 conjugacy classes of subgroups — too many to draw. Only its 134 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)(11,17)(12,20)(15,23)(16,18)(21,24);(2,5)(6,8)(10,15)(12,20)(14,23)(16,21)(18,24)(19,22);(9,10)(11,18)(12,21)(13,16)(14,17)(15,20)(19,23)(22,24);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^9|d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,f^a=fi,g^a=ghi,c^b=cg,f^b=fh,g^b=gi,f^c=fi,g^c=gi,h^c=hi,g^f=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72271403 |
| maximal normal subgroup of | #72271403 |
| maximal quotient of | #72271421 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406467);