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 | |
2175 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)(3,7)(4,6)(5,8)(9,10)(11,20)(12,26)(13,17)(14,19)(15,22)(16,30)(18,28)(21,33)(23,24)(25,35)(27,37)(29,36)(31,34)(32,40)(38,39);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,27)(13,22)(14,24)(16,33)(17,20)(18,23)(19,35)(21,31)(25,28)(26,32)(29,40)(30,38)(34,39)(36,37);(9,12,28,31,13,29,14,30)(10,16,19,36,17,34,18,26)(11,21,25,40,22,38,24,27)(15,32,35,33,20,37,23,39);(2,6)(5,7)(10,17)(11,22)(16,34)(18,19)(21,38)(24,25)(26,36)(27,40);(3,8)(5,7)(9,10)(11,23)(12,16)(13,17)(14,18)(15,25)(19,28)(20,24)(21,39)(22,35)(26,30)(27,37)(29,34)(31,36)(32,40)(33,38)Polycyclic presentation:
2^9|c^2=gi,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, 6589856);