Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806671 |
| 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 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2176 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6)(7,10)(8,12)(9,11)(13,14)(15,20)(16,24)(17,19)(18,21)(22,25)(23,27)(26,28);(5,7,10,6)(8,12,9,11)(14,19)(15,22)(16,25)(18,23)(20,26)(24,28);(2,4)(13,15,23,28,17,24,18,22)(14,20,27,26,19,16,21,25);(5,8,10,9)(6,11,7,12)(14,19)(15,24)(21,27)(22,28);(5,9,10,8)(6,11,7,12)(13,16,18,26,17,20,23,25)(14,15,21,22,19,24,27,28)Polycyclic presentation:
2^9|b^2=fi,c^2=hi,d^2=fi,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6718586);