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 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1979 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)(3,8)(4,6)(5,7)(9,10)(11,15)(12,17)(13,19)(14,20)(16,27)(18,30)(21,37)(22,31)(23,24)(25,32)(26,33)(28,35)(29,36)(34,38)(39,40);(1,3,5,4)(2,6,7,8)(9,11)(10,15)(12,22)(13,24)(14,26)(16,34)(17,31)(18,32)(19,23)(20,33)(21,28)(25,30)(27,38)(29,39)(35,37)(36,40);(9,12,13,27,14,29,30,28)(10,16,18,17,20,35,19,36)(11,21,24,22,26,38,25,39)(15,31,32,37,33,40,23,34);(1,4,5,3)(2,6,7,8)(9,12)(10,17)(11,22)(13,27)(14,29)(15,31)(16,19)(18,35)(20,36)(21,25)(23,34)(24,38)(26,39)(28,30)(32,37)(33,40);(9,10)(11,23)(12,16)(13,18)(14,20)(15,24)(17,27)(19,30)(21,34)(22,31)(25,33)(26,32)(28,36)(29,35)(37,38)(39,40)Polycyclic presentation:
2^9|b^2=f,c^2=g,d^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, 6491510);