Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #16249 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C23×C4 | C24 |
| commutator, socle | C24 | C23×C4 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1152 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,11)(8,9)(10,12)(13,14)(15,19)(16,18)(17,20)(21,22,26,23,25,28,34,30)(24,27,33,29,32,35,36,31);(6,9)(10,11)(13,15)(14,17)(16,20)(18,19)(21,23,26,28,25,30,34,22)(24,31,33,27,32,29,36,35);(2,4)(15,20)(17,19)(21,24)(22,27)(23,29)(25,32)(26,33)(28,35)(30,31)(34,36);(5,7,8,12)(6,10,9,11)(14,18)(17,19)(24,32)(27,35)(29,31)(33,36)Polycyclic presentation:
2^9|a^2=h,b^2=h,d^2=eg,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,d^c=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 278292);