Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574077 |
| 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 | |
1777 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)(9,10,15,14)(11,22,26,17)(12,29,13,19)(16,21,34,20)(18,32,30,31)(23,36,40,39)(24,37,33,35)(25,28,38,27);(2,6)(7,8)(9,11)(10,17)(12,24)(13,33)(14,22)(15,26)(16,27)(18,36)(19,37)(20,25)(21,38)(23,32)(28,34)(29,35)(30,39)(31,40);(9,12,16,31,15,13,34,32)(10,18,20,29,14,30,21,19)(11,23,27,24,26,40,28,33)(17,35,25,39,22,37,38,36);(1,3,5,4)(2,7,6,8)(9,13,15,12)(10,19,14,29)(11,24,26,33)(16,32,34,31)(17,35,22,37)(18,21,30,20)(23,27,40,28)(25,39,38,36);(1,4,5,3)(2,8,6,7)(9,14,15,10)(11,25,26,38)(12,30,13,18)(16,21,34,20)(17,27,22,28)(19,32,29,31)(23,39,40,36)(24,37,33,35)Polycyclic presentation:
2^9|a^2=i,c^2=g,d^2=f,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6492253);