Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667556 |
| 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 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2029 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)(5,6,9,11)(7,12,8,10)(14,18)(15,22)(19,21)(23,28);(5,6)(7,10)(8,12)(9,11)(13,14,16,18)(15,20,22,25)(17,19,24,21)(23,26,28,27);(1,2,3,4)(14,19)(15,22)(17,24)(18,21)(20,27)(25,26);(5,7,9,8)(6,10,11,12)(13,15,16,22)(14,20,18,25)(17,23,24,28)(19,26,21,27);(5,8,9,7)(6,10,11,12)(14,21)(17,24)(18,19)(20,27)(23,28)(25,26)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=g,d^2=f,e^2=fi,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, 6593945);