Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #125068 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#99 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×C22⋊C4 | C23 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1050 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 1050 conjugacy classes of subgroups — too many to draw. Only its 110 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2)(3,4)(5,6,8,11,9,12,7,10)(13,14)(15,21,23,17)(16,19)(18,20)(22,27,28,24)(25,26);(6,10,12,11)(13,15,17,20)(14,18,23,21)(16,22,24,26)(19,25,28,27);(2,4)(13,16,18,25)(14,19,20,26)(15,22,23,28)(17,24,21,27)Polycyclic presentation:
2^9|a^2=f,b^2=dfh,c^2=i,d^2=ghi,f^2=gh,b^a=bd,c^a=ce,d^a=dg,g^a=gi,h^a=hi,d^b=dh,f^b=fhi,g^b=gi,h^b=hi,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72338272, #72338277, #72338291, … |
| maximal normal subgroup of | #72338272, #72338277, #72338291, … |
| maximal quotient of | #72338358, #72338381 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 45465);