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 | |
1796 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)(9,10,14,13)(11,21,25,16)(12,18,29,32)(15,20,33,19)(17,31,28,30)(22,35,39,38)(23,37,40,34)(24,27,36,26);(2,6)(7,8)(9,11)(10,16)(12,23)(13,21)(14,25)(15,26)(17,35)(18,34)(19,24)(20,36)(22,31)(27,33)(28,38)(29,40)(30,39)(32,37);(9,12,15,30,14,29,33,31)(10,17,19,18,13,28,20,32)(11,22,26,23,25,39,27,40)(16,34,24,38,21,37,36,35);(1,3,5,4)(2,7,6,8)(9,12)(10,18)(11,23)(13,32)(14,29)(15,30)(16,34)(17,20)(19,28)(21,37)(22,27)(24,38)(25,40)(26,39)(31,33)(35,36);(1,4,5,3)(2,8,6,7)(9,13,14,10)(11,24,25,36)(12,28,29,17)(15,20,33,19)(16,26,21,27)(18,30,32,31)(22,38,39,35)(23,37,40,34)Polycyclic presentation:
2^9|a^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, 6492243);