Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #156684 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C25 | C24 |
| LCS | G ▷ C25 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C1 | |
3136 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 24 points:
(5,6)(7,8)(10,11)(17,18,21,22)(19,23,24,20);(2,4)(5,6,7,8)(14,16)(19,24)(20,23);(9,10,12,11)(13,14,15,16)(17,19)(18,20)(21,24)(22,23);(1,2)(3,4)(5,6)(7,8)(9,11,12,10)(17,20)(18,19)(21,23)(22,24)Polycyclic presentation:
2^9|a^2=fh,b^2=e,c^2=fg,d^2=h,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 65689);