Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #45332 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2026 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(5,6)(7,9)(8,10)(11,12)(13,14,17,21)(15,23,24,19)(16,20,26,25)(18,22,27,28);(1,2,4,3)(5,7)(6,9)(8,11)(10,12)(13,15,17,24)(14,19,21,23)(16,22,26,28)(18,25,27,20);(6,10)(7,11)(13,14)(15,23)(16,25)(17,21)(18,22)(19,24)(20,26)(27,28);(6,10)(9,12)(13,16,18,23,17,26,27,19)(14,20,22,24,21,25,28,15);(1,3,4,2)(14,21)(16,26)(19,23)(22,28)Polycyclic presentation:
2^9|a^2=i,b^2=f,d^2=h,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7518770);