Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417589 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
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 | |
1018 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10,12,13)(11,14,15,16)(17,18,20,22)(19,21,23,24)(25,26,28,32)(27,34,35,30)(29,31,37,33)(36,39,40,38);(1,3)(2,6)(4,7)(5,8)(17,19)(18,21)(20,23)(22,24)(25,27,28,35)(26,30,32,34)(29,36,37,40)(31,38,33,39);(1,4,5,2)(3,7,8,6)(10,13)(14,16)(19,23)(21,24);(3,8)(6,7)(9,11,12,15)(10,14,13,16)(26,31)(29,37)(30,38)(32,33)(34,39)(36,40)Polycyclic presentation:
2^9|a^2=efg,b^2=i,c^2=ei,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, 343415);