Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573635 |
| 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 | |
1963 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,6)(3,7,4,8)(9,10)(11,15)(12,25)(13,18)(14,19)(16,28)(17,31)(20,35)(21,22)(23,33)(24,34)(26,40)(27,32)(29,37)(30,38)(36,39);(2,6)(7,8)(9,11)(10,15)(12,26)(13,22)(14,24)(16,35)(17,33)(18,21)(19,34)(20,28)(23,31)(25,40)(27,29)(30,39)(32,37)(36,38);(9,12,14,29)(10,16,19,38)(11,20,24,39)(13,30,31,28)(15,32,34,40)(17,37,18,25)(21,35,33,36)(22,26,23,27);(1,3,5,4)(2,7,6,8)(9,11,14,24)(10,15,19,34)(12,27,29,26)(13,22,31,23)(16,36,38,35)(17,33,18,21)(20,28,39,30)(25,32,37,40);(1,4,5,3)(2,8,6,7)(9,10)(11,21)(12,16)(13,17)(14,19)(15,22)(18,31)(20,35)(23,34)(24,33)(25,28)(26,32)(27,40)(29,38)(30,37)(36,39)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=fi,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=dh,g^c=gi,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, 6485602);