Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7606 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C2×C22⋊C4 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C22×C42 | C23 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C23 ▷ C1 | |
808 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2,4,3)(5,6)(7,9)(8,11)(10,12)(13,14,16,18)(15,19,20,17);(6,9,11,10)(13,15)(14,17,18,19)(16,20)(21,22,23,26,24,25,27,28);(1,3,4,2)(15,20)(17,19)(22,25)(26,28)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=eh,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 | #72217683, #72217734 |
| maximal normal subgroup of | #72217683, #72217734 |
| maximal quotient of | #72213034, #72213038, #72213052 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3936);