Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| 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 | |
2181 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)(15,23)(19,27)(24,28)(25,26);(5,6)(7,10)(8,9)(11,12)(13,14,17,22)(15,24,23,28)(16,20,18,21)(19,26,27,25);(1,2,3,4)(6,9)(10,11)(13,15)(14,19)(16,25)(17,23)(18,26)(20,28)(21,24)(22,27);(5,7,8,12)(6,10,9,11)(13,16,17,18)(14,20,22,21)(15,25,23,26)(19,28,27,24);(14,21)(16,18)(19,24)(20,22)(25,26)(27,28)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fgi,d^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6592206);