Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519075 |
| 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 | |
2069 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)(12,25)(13,23)(14,19)(15,21)(17,29)(18,32)(20,27)(22,31)(24,33)(26,34)(28,37)(30,35)(36,38)(39,40);(2,6)(7,8)(9,11,15,18)(10,16,21,32)(12,26,30,39)(13,20,33,19)(14,23,27,24)(17,31,36,28)(22,38,37,29)(25,34,35,40);(1,3,5,4)(2,7,6,8)(9,12,27,38,15,30,14,29)(10,17,19,35,21,36,20,25)(11,22,23,39,18,37,24,26)(13,31,16,34,33,28,32,40);(1,4,5,3)(2,8,6,7)(9,13,27,16,15,33,14,32)(10,18,19,24,21,11,20,23)(12,28,38,40,30,31,29,34)(17,22,35,39,36,37,25,26);(2,6)(7,8)(9,14,15,27)(10,19,21,20)(11,23,18,24)(12,29,30,38)(13,32,33,16)(17,35,36,25)(22,39,37,26)(28,34,31,40)Polycyclic presentation:
2^9|b^2=i,c^2=fgi,d^2=fgi,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,d^c=di,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7469010);