Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514172 |
| 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 |
| prime radical | #61276 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1713 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)(3,8)(4,6)(5,7)(9,10)(11,17)(12,24)(13,26)(14,18)(15,30)(16,19)(20,21)(22,35)(23,37)(25,33)(27,28)(29,32)(31,36)(34,39)(38,40);(1,3,5,4)(2,6,7,8)(9,11,16,23)(10,17,19,37)(12,25,27,34)(13,21,29,40)(14,22,15,31)(18,35,30,36)(20,32,38,26)(24,33,28,39);(1,4,5,3)(2,6,7,8)(9,12,16,27)(10,13,19,29)(11,20,23,38)(14,26,15,32)(17,33,37,39)(18,28,30,24)(21,36,40,35)(22,34,31,25);(9,13,16,29)(10,12,19,27)(11,21,23,40)(14,24,15,28)(17,34,37,25)(18,32,30,26)(20,36,38,35)(22,33,31,39);(2,7)(6,8)(9,14,16,15)(10,18,19,30)(11,22,23,31)(12,26,27,32)(13,28,29,24)(17,35,37,36)(20,34,38,25)(21,39,40,33)Polycyclic presentation:
2^9|b^2=fi,c^2=fi,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,g^c=gi,h^c=hi,e^d=ei,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7422671);