Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #129414 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#99 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C23×Q8 | C23 |
| prime radical | C24 | C2×C22⋊C4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×C4×Q8 ◁ G | |
| LECS | G ▷ C23×Q8 ▷ C22 ▷ C2 ▷ C1 | |
960 conjugacy classes of subgroups; only the 154 normal subgroups are shown.
This group has 960 conjugacy classes of subgroups — too many to draw. Only its 154 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) = 128 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2)(3,4)(13,14,18,23,17,22,15,20)(16,19,24,27,25,21,26,28);(5,6,8,10)(7,11,12,9)(14,19)(15,18)(16,25)(20,28)(21,22)(23,27);(2,4)(5,7,8,12)(6,9,10,11)(13,15,17,18)(14,20,22,23)(16,24,25,26)(19,27,21,28)Polycyclic presentation:
2^9|a^2=g,b^2=fi,c^2=f,d^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,c^b=cf,d^b=di,g^b=ghi,g^d=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72268166, #72268168, #72268169 |
| maximal normal subgroup of | #72268166, #72268168, #72268169 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56628);