Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519063 |
| 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 | |
2118 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)(3,7)(4,8)(5,6)(9,10)(11,20)(12,26)(13,14)(15,19)(16,24)(17,30)(18,28)(21,37)(22,25)(23,33)(27,36)(29,31)(32,39)(34,40)(35,38);(2,6)(7,8)(9,11,15,24)(10,16,19,20)(12,27,31,39)(13,22,18,33)(14,23,28,25)(17,34,35,37)(21,38,40,30)(26,32,29,36);(1,3,5,4)(2,7,6,8)(9,12,28,38,15,31,14,30)(10,17,13,29,19,35,18,26)(11,21,23,39,24,40,25,27)(16,32,33,37,20,36,22,34);(1,4,5,3)(2,8,6,7)(9,13,28,19,15,18,14,10)(11,22,23,16,24,33,25,20)(12,29,38,35,31,26,30,17)(21,34,39,32,40,37,27,36);(2,6)(7,8)(9,14,15,28)(10,18,19,13)(11,23,24,25)(12,30,31,38)(16,33,20,22)(17,26,35,29)(21,39,40,27)(32,37,36,34)Polycyclic presentation:
2^9|b^2=i,c^2=fg,d^2=fg,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,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7468972);