Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7021375 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2111 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)(3,5)(4,6)(7,8)(9,10,14,18,15,20,26,19)(11,21,24,28,25,12,23,16)(13,17,30,34,31,36,39,35)(22,33,29,40,38,27,37,32);(1,2)(3,7)(4,6)(5,8)(9,11)(10,16)(12,18)(13,29)(14,23)(15,25)(17,33)(19,21)(20,28)(22,30)(24,26)(27,36)(31,37)(32,34)(35,40)(38,39);(1,3)(2,5)(4,8)(6,7)(9,12,26,28,15,21,14,16)(10,11,18,24,20,25,19,23)(13,27,39,40,31,33,30,32)(17,22,34,29,36,38,35,37);(3,8)(5,7)(9,13,14,30,15,31,26,39)(10,17,18,34,20,36,19,35)(11,22,24,29,25,38,23,37)(12,27,16,32,21,33,28,40);(2,6)(5,7)(9,14,15,26)(10,18,20,19)(11,23,25,24)(12,28,21,16)(13,30,31,39)(17,34,36,35)(22,37,38,29)(27,40,33,32)Polycyclic presentation:
2^9|a^2=f,c^2=gi,d^2=f,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6928576);