Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907849 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #30105 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1525 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(9,10)(11,17)(12,16)(13,15)(14,19)(18,24)(20,22)(21,23)(25,26,28,30)(27,33,34,40)(29,37,35,32)(31,39,38,36);(1,2)(3,7)(4,6)(5,8)(9,11,13,19)(10,14,15,17)(12,20,21,18)(16,24,23,22)(26,30)(27,34)(32,37)(36,39);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,16)(13,15)(17,19)(18,22)(20,24)(21,23)(25,27,35,39,28,34,29,36)(26,31,37,40,30,38,32,33);(3,8)(5,7)(10,15)(14,17)(16,23)(22,24)(26,30)(31,38)(32,37)(33,40);(2,6)(5,7)(9,12,13,21)(10,16,15,23)(11,18,19,20)(14,22,17,24)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=gi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=dfh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6820034);