Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500908 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1909 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(9,10,14,19,15,12,25,18)(11,16,23,33,24,26,22,20)(13,28,30,27,31,34,29,17)(21,35,36,39,37,40,38,32);(1,3,5,2)(4,7,8,6)(9,11,15,24)(10,16,12,26)(13,21,31,37)(14,23,25,22)(17,32,27,39)(18,20,19,33)(28,35,34,40)(29,38,30,36);(1,4,5,8)(2,6,3,7)(9,12)(10,15)(11,20)(13,27)(14,19)(16,22)(17,31)(18,25)(21,35)(23,26)(24,33)(28,30)(29,34)(32,36)(37,40)(38,39);(2,3)(6,7)(9,13)(10,17)(11,21)(12,27)(14,29)(15,31)(16,32)(18,28)(19,34)(20,35)(22,36)(23,38)(24,37)(25,30)(26,39)(33,40);(9,14,15,25)(10,18,12,19)(11,22,24,23)(13,29,31,30)(16,33,26,20)(17,28,27,34)(21,36,37,38)(32,40,39,35)Polycyclic presentation:
2^9|a^2=g,b^2=fi,c^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,e^b=ei,h^b=hi,g^c=gi,h^c=hi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7410872);