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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2023 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,10)(8,9)(11,12)(13,14,19,23)(15,24,26,16)(17,22,18,28)(20,21,27,25);(2,4)(6,9)(10,12)(13,14,19,23)(15,24,26,16)(17,22,18,28)(20,21,27,25);(5,7,8,11)(6,10,9,12)(13,15,17,21,19,26,18,25)(14,20,28,16,23,27,22,24);(1,2,3,4)(13,16,17,27,19,24,18,20)(14,21,28,26,23,25,22,15);(6,9)(10,12)(13,17,19,18)(14,22,23,28)(15,21,26,25)(16,20,24,27)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fgi,d^2=hi,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,d^c=di,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7520928);