Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518992 |
| 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 | |
2151 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)(15,25)(16,21)(17,22)(18,23)(19,24)(20,26)(27,28);(1,2,4,3)(5,7)(6,9)(8,11)(10,12)(13,15,18,26)(14,20,23,25)(16,17,19,27)(21,28,24,22);(6,10)(7,11)(13,16,18,19)(14,21,23,24)(15,17,26,27)(20,28,25,22);(6,10)(9,12)(13,17,19,26,18,27,16,15)(14,22,24,20,23,28,21,25);(1,3,4,2)(14,23)(20,25)(21,24)(22,28)Polycyclic presentation:
2^9|b^2=f,c^2=i,d^2=h,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7518683);