Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417589 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1236 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,8,9)(7,10,11,12)(13,14,16,18)(15,17,19,20)(21,22)(23,31)(24,27)(25,29)(26,28)(30,34)(32,36)(33,35);(1,2,3,4)(13,15)(14,17)(16,19)(18,20)(21,23,25,24)(22,27,29,31)(26,32,34,33)(28,35,30,36);(6,9)(10,12)(15,19)(17,20)(21,24,25,23)(22,27,29,31)(26,33,34,32)(28,35,30,36);(2,4)(5,7,8,11)(6,10,9,12)(22,28)(26,34)(27,35)(29,30)(31,36)(32,33)Polycyclic presentation:
2^9|a^2=fg,b^2=e,c^2=i,d^2=f,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=dei,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 343276);