Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7501009 |
| 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×#3041 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1996 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)(9,10,14,20,15,21,13,19)(11,16,23,32,24,33,30,18)(12,25,27,36,29,35,28,17)(22,34,26,39,38,40,37,31);(1,3,5,2)(4,7,8,6)(9,11)(10,16)(12,26)(13,30)(14,23)(15,24)(17,34)(18,19)(20,32)(21,33)(22,28)(25,39)(27,38)(29,37)(31,35)(36,40);(1,4,5,8)(2,6,3,7)(9,12)(10,17)(11,22)(13,27)(14,28)(15,29)(16,31)(18,34)(19,25)(20,35)(21,36)(23,37)(24,38)(26,30)(32,40)(33,39);(2,3)(6,7)(9,11)(10,18)(12,22)(13,23)(14,30)(15,24)(16,19)(17,34)(20,33)(21,32)(25,31)(26,28)(27,37)(29,38)(35,39)(36,40);(9,13,15,14)(10,19,21,20)(11,23,24,30)(12,27,29,28)(16,32,33,18)(17,25,36,35)(22,37,38,26)(31,40,39,34)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,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, 7410914);