Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #16249 |
| 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 | |
1150 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,10)(8,9)(11,12)(13,14,17,16)(15,19,20,18);(1,2)(3,4)(13,15,17,20)(14,18,16,19)(21,22)(23,26)(24,28)(25,27)(29,33)(30,32)(31,34)(35,36);(2,4)(6,9)(10,12)(14,16)(18,19)(21,23,24,30,25,29,31,35)(22,26,28,32,27,33,34,36);(5,7,8,11)(6,10,9,12)(13,16,17,14)(15,19,20,18)(22,27)(23,29)(28,34)(30,35)Polycyclic presentation:
2^9|a^2=e,b^2=e,c^2=h,d^2=f,h^2=i,b^a=be,c^a=cf,c^b=cg,d^b=di,d^c=dei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290069);