Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #36644 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2582 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)(7,10)(8,11)(9,12)(13,14,16,20,17,19,23,26)(15,18,22,25,21,24,27,28);(5,7,10,6)(8,11,9,12)(13,15,16,22,17,21,23,27)(14,18,20,25,19,24,26,28);(1,2,3,4)(5,8,10,9)(6,11,7,12);(5,9,10,8)(6,11,7,12)(15,21)(18,24)(22,27)(25,28);(14,19)(18,24)(20,26)(25,28)Polycyclic presentation:
2^9|a^2=h,b^2=fh,c^2=fg,d^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=cf,d^b=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6283599);