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 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2150 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)(3,7)(4,6)(5,8)(9,10,16,11)(12,23,26,32)(13,18,15,20)(14,19,30,22)(17,31,21,24)(25,36,39,40)(27,34,29,28)(33,38,35,37);(1,3)(2,5)(4,8)(6,7)(9,11,16,10)(12,23,26,32)(13,18,15,20)(14,28,30,34)(17,24,21,31)(19,27,22,29)(25,35,39,33)(36,37,40,38);(9,12,15,31,16,26,13,24)(10,17,18,32,11,21,20,23)(14,25,29,38,30,39,27,37)(19,33,34,40,22,35,28,36);(2,6)(5,7)(9,13,16,15)(10,18,11,20)(12,24,26,31)(14,29,30,27)(17,32,21,23)(19,28,22,34)(25,38,39,37)(33,36,35,40);(3,8)(5,7)(9,14)(10,19)(11,22)(12,25)(13,27)(15,29)(16,30)(17,33)(18,34)(20,28)(21,35)(23,36)(24,37)(26,39)(31,38)(32,40)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=gi,d^2=i,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,h^c=hi,e^d=ei,h^e=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6589863);