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 | |
2624 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)(4,6)(5,8)(9,10)(11,20)(12,25)(13,18)(14,30)(15,19)(16,23)(17,28)(21,34)(22,31)(24,33)(26,37)(27,36)(29,35)(32,40)(38,39);(1,3)(2,5)(4,8)(6,7)(9,11,13,23)(10,16,18,20)(12,26,27,40)(14,24,15,31)(17,34,35,38)(19,33,30,22)(21,28,39,29)(25,32,36,37);(9,12,13,27)(10,17,18,35)(11,21,23,39)(14,29,15,28)(16,32,20,37)(19,36,30,25)(22,38,33,34)(24,40,31,26);(2,6)(5,7)(10,18)(16,20)(17,35)(19,30)(22,33)(25,36)(32,37)(34,38);(3,8)(5,7)(9,10)(11,22)(12,17)(13,18)(14,19)(15,30)(16,24)(20,31)(21,38)(23,33)(25,28)(26,37)(27,35)(29,36)(32,40)(34,39)Polycyclic presentation:
2^9|b^2=i,c^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, 6575172);