Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682638 |
| 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 | |
1791 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)(3,6)(4,5)(7,8)(9,10,15,20)(11,17,23,16)(12,25,24,32)(13,21,26,18)(14,19,29,36)(22,33,38,31)(27,40,39,35)(28,37,30,34);(2,5)(3,7)(9,11)(10,16)(12,26)(13,24)(14,27)(15,23)(17,20)(18,32)(19,35)(21,25)(22,28)(29,39)(30,38)(31,37)(33,34)(36,40);(9,12,13,11,15,24,26,23)(10,17,18,25,20,16,21,32)(14,27,28,22,29,39,30,38)(19,33,34,40,36,31,37,35);(2,5)(6,8)(9,13,15,26)(10,18,20,21)(11,12,23,24)(14,28,29,30)(16,25,17,32)(19,34,36,37)(22,27,38,39)(31,40,33,35);(1,3,4,7)(2,6,5,8)(9,14,15,29)(10,19,20,36)(11,22,23,38)(12,27,24,39)(13,28,26,30)(16,31,17,33)(18,34,21,37)(25,40,32,35)Polycyclic presentation:
2^9|a^2=i,c^2=g,d^2=i,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6592723);