Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7606 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C2×C22⋊C4 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C22×C42 | C23 |
| prime radical | C25 | C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C23 ▷ C1 | |
1434 conjugacy classes of subgroups; only the 234 normal subgroups are shown.
This group has 1434 conjugacy classes of subgroups — too many to draw. Only its 234 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) = 256 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2,3,4)(5,6)(7,11)(8,10)(9,12)(13,14)(15,19)(16,18)(17,20);(6,10,12,11)(13,15)(14,17,18,19)(16,20)(21,22,23,26,24,25,27,28);(5,7,9,8)(6,11,12,10)(15,20)(17,19)(22,25)(26,28)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=g,d^2=g,f^2=i,b^a=bd,d^a=dg,f^a=fgh,c^b=chi,d^b=dh104 conjugacy classes — view the full character table.
| maximal subgroup of | #72217697, #72227644, #72227647, … |
| maximal normal subgroup of | #72217697, #72227644, #72227647, … |
| maximal quotient of | #72213034, #72213056, #72227993, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3451);