Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996799 |
| 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 | |
2255 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)(9,10,13,18,15,19,14,29)(11,20,23,32,24,31,22,16)(12,17,27,35,28,36,26,34)(21,33,25,39,37,40,38,30);(1,2)(3,7)(4,6)(5,8)(9,11,15,24)(10,16,19,32)(12,25,28,38)(13,22,14,23)(17,33,36,40)(18,31,29,20)(21,26,37,27)(30,34,39,35);(1,3)(2,5)(4,8)(6,7)(9,10,15,19)(11,16,24,32)(12,17,28,36)(13,18,14,29)(20,22,31,23)(21,30,37,39)(25,33,38,40)(26,34,27,35);(3,8)(5,7)(9,12,13,27,15,28,14,26)(10,17,18,35,19,36,29,34)(11,21,23,25,24,37,22,38)(16,30,20,33,32,39,31,40);(2,6)(5,7)(9,13,15,14)(10,18,19,29)(11,22,24,23)(12,26,28,27)(16,31,32,20)(17,34,36,35)(21,25,37,38)(30,33,39,40)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,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,g^c=gi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6904008);