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 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1702 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)(11,18)(12,26)(13,28)(14,22)(15,21)(16,20)(17,25)(19,31)(23,37)(24,33)(27,32)(29,36)(30,35)(34,38)(39,40);(2,6)(7,8)(9,11,15,25)(10,17,21,18)(12,14,24,16)(13,23,30,39)(19,34,36,32)(20,33,22,26)(27,29,38,31)(28,40,35,37);(1,3,5,4)(2,7,6,8)(9,12,16,11,15,24,14,25)(10,18,22,26,21,17,20,33)(13,27,31,23,30,38,29,39)(19,34,28,37,36,32,35,40);(9,13,14,29,15,30,16,31)(10,19,20,35,21,36,22,28)(11,23,12,27,25,39,24,38)(17,32,26,37,18,34,33,40);(1,4,5,3)(2,7,6,8)(9,14,15,16)(10,20,21,22)(11,24,25,12)(13,29,30,31)(17,33,18,26)(19,35,36,28)(23,38,39,27)(32,40,34,37)Polycyclic presentation:
2^9|b^2=i,c^2=fgi,d^2=gi,e^2=f,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,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, 7471404);