Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667556 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
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 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2146 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,8,9)(7,11,12,10)(14,19)(15,16)(22,27)(24,25);(5,6)(7,10)(8,9)(11,12)(13,14,17,19)(15,23,16,21)(18,27,26,22)(20,25,28,24);(1,2,3,4)(6,9)(10,11)(13,15)(14,20)(16,17)(18,25)(19,28)(21,22)(23,27)(24,26);(5,7,8,12)(6,10,9,11)(13,16,17,15)(14,21,19,23)(18,25,26,24)(20,27,28,22);(14,22)(18,26)(19,27)(20,21)(23,28)(24,25)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=fg,d^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6591920);