Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667964 |
| 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 | |
2649 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,16)(12,26)(13,18)(14,19)(15,21)(17,29)(20,31)(22,37)(23,34)(24,35)(25,36)(27,40)(28,39)(30,38)(32,33);(1,3)(2,5)(4,8)(6,7)(9,11,15,25)(10,16,21,36)(12,27,30,32)(13,23,31,24)(14,29,22,28)(17,37,39,19)(18,34,20,35)(26,40,38,33);(9,12)(10,17)(11,22)(13,28)(14,25)(15,30)(16,33)(18,26)(19,35)(20,38)(21,39)(23,27)(24,32)(29,31)(34,37)(36,40);(2,6)(5,7)(9,13,15,31)(10,18,21,20)(11,23,25,24)(12,28,30,29)(14,32,22,27)(16,34,36,35)(17,26,39,38)(19,33,37,40);(3,8)(5,7)(9,14,15,22)(10,19,21,37)(11,12,25,30)(13,27,31,32)(16,26,36,38)(17,35,39,34)(18,40,20,33)(23,29,24,28)Polycyclic presentation:
2^9|b^2=i,d^2=i,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,g^c=gi,h^c=hi,e^d=ei,h^e=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6575171);