Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682648 |
| 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 | |
2152 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)(11,21)(12,26)(13,20)(14,19)(15,18)(16,25)(17,30)(22,38)(23,33)(24,31)(27,32)(28,36)(29,35)(34,40)(37,39);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,27)(13,31)(14,25)(15,23)(17,34)(18,24)(19,21)(20,33)(22,30)(26,37)(28,40)(29,39)(32,35)(36,38);(9,12,15,30,14,29,13,28)(10,17,20,26,19,36,18,35)(11,22,23,39,25,40,31,27)(16,32,33,34,21,37,24,38);(2,6)(5,7)(9,13,14,15)(10,18,19,20)(11,23,25,31)(12,28,29,30)(16,33,21,24)(17,35,36,26)(22,39,40,27)(32,34,37,38);(3,8)(5,7)(9,10)(11,24)(12,17)(13,18)(14,19)(15,20)(16,23)(21,31)(22,38)(25,33)(26,30)(27,37)(28,35)(29,36)(32,39)(34,40)Polycyclic presentation:
2^9|c^2=g,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,h^e=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6589848);