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 | |
1758 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)(3,6)(4,5)(7,8)(9,10)(11,20)(12,24)(13,26)(14,31)(15,19)(16,23)(17,29)(18,32)(21,37)(22,34)(25,35)(27,38)(28,33)(30,40)(36,39);(2,5)(6,8)(9,11,15,23)(10,16,19,20)(12,14,22,32)(13,27,30,21)(17,33,36,35)(18,34,31,24)(25,39,28,29)(26,37,40,38);(1,3,4,7)(2,6,5,8)(9,12,14,23,15,22,32,11)(10,16,18,24,19,20,31,34)(13,28,29,21,30,25,39,27)(17,35,26,37,36,33,40,38);(9,13,14,29,15,30,32,39)(10,17,18,26,19,36,31,40)(11,21,12,25,23,27,22,28)(16,33,24,38,20,35,34,37);(2,5)(6,8)(9,14,15,32)(10,18,19,31)(11,22,23,12)(13,29,30,39)(16,34,20,24)(17,26,36,40)(21,28,27,25)(33,37,35,38)Polycyclic presentation:
2^9|b^2=i,c^2=fgi,d^2=g,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,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, 7467936);