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)(11,17)(12,25)(13,19)(14,20)(15,32)(16,21)(18,27)(22,34)(23,35)(24,36)(26,33)(28,39)(29,38)(30,40)(31,37);(1,3)(2,5)(4,8)(6,7)(9,11)(10,17)(12,26)(13,22)(14,28)(15,23)(16,24)(18,37)(19,34)(20,39)(21,36)(25,33)(27,31)(29,30)(32,35)(38,40);(9,12,16,29)(10,18,21,39)(11,14,24,31)(13,27,15,28)(17,33,36,40)(19,38,32,25)(20,34,37,35)(22,26,23,30);(2,6)(5,7)(9,13,16,15)(10,19,21,32)(11,22,24,23)(12,27,29,28)(14,26,31,30)(17,34,36,35)(18,38,39,25)(20,33,37,40);(3,8)(5,7)(9,14,16,31)(10,20,21,37)(11,12,24,29)(13,30,15,26)(17,25,36,38)(18,34,39,35)(19,40,32,33)(22,28,23,27)Polycyclic presentation:
2^9|c^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, 6575167);