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 | |
1778 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)(10,17)(12,16)(13,23)(14,24)(15,25)(18,21)(19,35)(20,34)(22,26)(27,32)(28,40)(29,30)(31,39)(33,37)(36,38);(1,3,4,7)(2,6,5,8)(9,12,14,25,15,24,16,11)(10,18,20,34,21,17,22,26)(13,29,30,23,31,27,32,39)(19,33,36,38,37,35,28,40);(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|c^2=fg,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, 7467926);