Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499916 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#447 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1095 conjugacy classes of subgroups; only the 156 normal subgroups are shown.
This group has 1095 conjugacy classes of subgroups — too many to draw. Only its 156 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)(12,21)(13,16)(15,19)(17,20)(18,22)(23,24);(1,3,4,8)(2,5,6,7)(11,18)(14,22)(15,23)(19,24);(10,14)(11,19)(12,20)(15,17)(16,22)(18,24)(21,23);(2,6)(5,7)Polycyclic presentation:
2^9|b^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,h^a=hi,c^b=cg,f^b=fi,f^c=fh,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72237276, #72239808, #72239816, … |
| maximal normal subgroup of | #72237276, #72239808, #72239816, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407136);