Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500736 |
| 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 | #60456 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2120 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,4)(5,7)(6,8)(9,10,14,19)(11,20,23,32)(12,25,28,34)(13,30,15,18)(16,22,33,24)(17,29,26,27)(21,36,39,31)(35,38,37,40);(1,3)(2,6)(4,7)(5,8)(9,10)(11,20)(12,26)(13,18)(14,19)(15,30)(16,22)(17,28)(21,37)(23,32)(24,33)(25,29)(27,34)(31,40)(35,39)(36,38);(1,4,5,6)(2,3,7,8)(9,11,14,23)(10,16,19,33)(12,21,28,39)(13,22,15,24)(17,31,26,36)(18,32,30,20)(25,37,34,35)(27,38,29,40);(2,7)(3,8)(9,12)(10,17)(11,21)(13,27)(14,28)(15,29)(16,31)(18,25)(19,26)(20,35)(22,38)(23,39)(24,40)(30,34)(32,37)(33,36);(9,13,14,15)(10,18,19,30)(11,22,23,24)(12,27,28,29)(16,32,33,20)(17,25,26,34)(21,38,39,40)(31,37,36,35)Polycyclic presentation:
2^9|a^2=i,c^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,g^a=gi,c^b=ch,d^b=df,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, 7408545);