Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682658 |
| 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 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1850 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,7)(3,8,4,6)(9,10,16,19)(11,17,24,20)(12,15,27,13)(14,18,31,36)(21,25,23,33)(22,32,39,37)(26,30,35,28)(29,34,38,40);(1,2)(3,6)(4,8)(5,7)(9,11)(10,17)(12,25)(13,21)(14,29)(15,23)(16,24)(18,34)(19,20)(22,30)(26,37)(27,33)(28,39)(31,38)(32,35)(36,40);(9,12,15,19,16,27,13,10)(11,20,23,33,24,17,21,25)(14,26,30,36,31,35,28,18)(22,37,38,40,39,32,29,34);(1,3,5,4)(2,6,7,8)(9,13,16,15)(10,12,19,27)(11,21,24,23)(14,28,31,30)(17,25,20,33)(18,26,36,35)(22,29,39,38)(32,34,37,40);(1,4,5,3)(2,6,7,8)(9,14)(10,18)(11,22)(12,26)(13,28)(15,30)(16,31)(17,32)(19,36)(20,37)(21,29)(23,38)(24,39)(25,34)(27,35)(33,40)Polycyclic presentation:
2^9|a^2=fi,c^2=g,d^2=fi,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,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, 6594192);