Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667460 |
| 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 ◁ C4×#124 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2125 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)(13,14,17,19)(15,23,25,20)(16,21,26,27)(18,22,24,28);(5,6)(7,10)(8,9)(11,12)(14,19)(15,16)(18,24)(20,21)(23,27)(25,26);(1,2,3,4)(6,9)(10,11)(13,15)(14,20)(16,24)(17,25)(18,26)(19,23)(21,22)(27,28);(5,7,8,12)(6,10,9,11)(14,19)(20,23)(21,27)(22,28);(13,16,17,26)(14,21,19,27)(15,24,25,18)(20,22,23,28)Polycyclic presentation:
2^9|a^2=f,c^2=fg,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,h^c=hi,h^e=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6591851);