Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806623 |
| 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 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1852 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(6,10)(7,8);(5,6)(7,12)(8,11)(9,10)(13,14)(15,16)(17,22)(18,23)(19,21)(20,25)(24,27)(26,28);(2,4)(5,7,9,8)(6,11,10,12)(13,15,16,14,19,23,18,21)(17,22,24,20,26,28,25,27);(5,8,9,7)(6,11,10,12)(13,16,19,18)(14,15,21,23)(17,24,26,25)(20,22,27,28);(13,17,18,25,19,26,16,24)(14,20,15,22,21,27,23,28)Polycyclic presentation:
2^9|c^2=hi,d^2=fi,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di74 conjugacy classes — view the full character table.
| maximal subgroup of | #72308331 |
| maximal normal subgroup of | #72308331 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6717810);