Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574084 |
| 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 | |
1758 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)(13,18)(14,20)(15,23)(19,30)(21,33)(22,34)(24,32)(25,29)(26,31)(27,35)(28,36)(37,40)(38,39);(2,6)(7,8)(9,10,13,18)(11,16,22,34)(12,25,27,36)(14,19,30,20)(15,23,32,24)(17,28,35,29)(21,31,37,39)(26,33,38,40);(9,11,14,23,13,22,30,24)(10,15,19,16,18,32,20,34)(12,21,28,26,27,37,29,38)(17,31,25,33,35,39,36,40);(1,3,5,4)(2,7,6,8)(9,11)(10,16)(12,26)(13,22)(14,23)(15,20)(17,31)(18,34)(19,32)(21,29)(24,30)(25,33)(27,38)(28,37)(35,39)(36,40);(1,4,5,3)(2,8,6,7)(9,12,13,27)(10,17,18,35)(11,21,22,37)(14,28,30,29)(15,31,32,39)(16,33,34,40)(19,25,20,36)(23,26,24,38)Polycyclic presentation:
2^9|b^2=i,c^2=g,d^2=fi,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,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, 6492244);