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 |
| prime radical | #61222 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1782 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)(11,16)(12,17)(13,19)(14,20)(15,22)(18,30)(21,33)(23,32)(24,34)(25,27)(26,31)(28,35)(29,36)(37,39)(38,40);(2,6)(7,8)(9,10)(11,16)(12,25)(13,18)(14,20)(15,23)(17,27)(19,30)(21,37)(22,32)(24,34)(26,40)(28,36)(29,35)(31,38)(33,39);(1,3,5,4)(2,7,6,8)(9,11,13,22,14,24,30,23)(10,15,18,16,20,32,19,34)(12,21,27,26,29,38,28,39)(17,31,35,33,36,37,25,40);(1,4,5,3)(2,8,6,7)(9,11)(10,16)(12,26)(13,22)(14,24)(15,19)(17,31)(18,32)(20,34)(21,28)(23,30)(25,40)(27,38)(29,39)(33,35)(36,37);(9,12,14,29)(10,17,20,36)(11,21,24,38)(13,27,30,28)(15,31,32,37)(16,33,34,40)(18,35,19,25)(22,26,23,39)Polycyclic presentation:
2^9|c^2=fg,d^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=ehi,h^b=hi,d^c=dh,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, 6492003);