Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514040 |
| 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 | |
1946 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)(7,10)(8,9)(11,12)(13,14)(15,19)(16,17)(18,20)(21,26)(22,28)(23,27)(24,25);(2,3)(6,9)(10,12)(13,15,18,23)(14,19,20,27)(16,21,25,28)(17,26,24,22);(1,2,4,3)(5,7,8,11)(6,10,9,12)(13,16,18,25)(14,17,20,24)(15,21,23,28)(19,26,27,22);(1,3,4,2)(13,14)(15,21)(16,24)(17,25)(18,20)(19,26)(22,27)(23,28);(6,9)(10,12)(14,20)(16,25)(19,27)(21,28)Polycyclic presentation:
2^9|b^2=i,c^2=fgi,d^2=g,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,e^d=ei,h^d=hi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7467855);