Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #17082 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C23×C4 | C24 |
| commutator, socle, prime radical | C24 | C23×C4 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1194 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(6,9)(7,11)(21,22,26,30,25,27,24,29)(23,28,32,35,31,34,33,36);(5,6,8,9)(7,12,11,10)(13,14,16,18)(15,19,20,17)(22,27)(23,31)(29,30)(32,33);(2,4)(13,15,16,20)(14,17,18,19)(21,23,25,31)(22,28,27,34)(24,33,26,32)(29,36,30,35);(2,4)(5,7,8,11)(6,10,9,12)(21,24,25,26)(22,29,27,30)(23,32,31,33)(28,35,34,36)Polycyclic presentation:
2^9|a^2=h,b^2=eg,c^2=g,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=dei,d^c=di92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 294295);