Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519083 |
| 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 | |
2053 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)(4,8)(5,6)(9,10)(11,16)(12,23)(13,21)(14,28)(15,19)(17,27)(18,32)(20,25)(22,34)(24,33)(26,38)(29,35)(30,31)(36,37)(39,40);(2,6)(7,8)(9,11,15,22)(10,16,19,34)(12,24,26,39)(13,28,30,18)(14,31,32,21)(17,25,36,29)(20,37,35,27)(23,33,38,40);(1,3,5,4)(2,7,6,8)(9,12,14,27,15,26,32,37)(10,17,18,23,19,36,28,38)(11,20,21,24,22,35,31,39)(13,29,16,33,30,25,34,40);(1,4,5,3)(2,8,6,7)(9,13,14,16,15,30,32,34)(10,11,18,21,19,22,28,31)(12,25,27,40,26,29,37,33)(17,35,23,39,36,20,38,24);(2,6)(7,8)(12,26)(17,36)(20,35)(23,38)(24,39)(25,29)(27,37)(33,40)Polycyclic presentation:
2^9|b^2=i,c^2=fg,d^2=fg,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,d^c=di,e^c=ei,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7469019);