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 | |
1766 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,18)(12,26)(13,28)(14,22)(15,21)(16,20)(17,25)(19,32)(23,38)(24,34)(27,33)(29,35)(30,37)(31,36)(39,40);(2,5)(6,8)(9,11,15,25)(10,17,21,18)(12,14,24,16)(13,23,31,39)(19,35,37,33)(20,34,22,26)(27,30,29,32)(28,40,36,38);(1,3,4,7)(2,6,5,8)(9,12,16,11,15,24,14,25)(10,18,22,26,21,17,20,34)(13,29,32,39,31,27,30,23)(19,33,28,40,37,35,36,38);(9,13,14,30,15,31,16,32)(10,19,20,36,21,37,22,28)(11,23,12,27,25,39,24,29)(17,33,26,38,18,35,34,40);(2,5)(6,8)(9,14,15,16)(10,20,21,22)(11,24,25,12)(13,30,31,32)(17,34,18,26)(19,36,37,28)(23,29,39,27)(33,40,35,38)Polycyclic presentation:
2^9|b^2=i,c^2=fgi,d^2=gi,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, 7467938);