Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667836 |
| 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 ◁ C4×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2117 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,11)(8,10)(9,12)(13,14,17,16)(15,21,24,19)(18,20,26,25)(22,28,23,27);(5,7,8,12)(6,9,10,11)(13,15)(14,19)(16,21)(17,24)(18,22)(20,27)(23,26)(25,28);(1,2,3,4)(15,22)(18,26)(19,27)(20,25)(21,28)(23,24);(6,10)(9,11)(13,16,17,14)(15,19,24,21)(18,25,26,20)(22,27,23,28);(7,12)(9,11)(15,23)(18,26)(19,28)(20,25)(21,27)(22,24)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=g,d^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,d^b=di,e^b=efh,h^b=hi,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6590570);