Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30490 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C2×D42 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2651 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)(9,10,14,20,16,22,15,21)(11,17,25,35,26,36,34,40)(12,27,29,37,30,24,28,18)(13,23,32,19,33,38,31,39);(1,3)(2,5)(4,8)(6,7)(9,11)(10,17)(12,19)(13,24)(14,25)(15,34)(16,26)(18,32)(20,35)(21,40)(22,36)(23,28)(27,33)(29,38)(30,39)(31,37);(9,12)(10,18)(11,23)(13,17)(14,28)(15,29)(16,30)(19,34)(20,24)(21,27)(22,37)(25,39)(26,38)(31,35)(32,40)(33,36);(2,6)(3,8)(9,13)(10,19)(11,24)(12,17)(14,31)(15,32)(16,33)(18,34)(20,23)(21,38)(22,39)(25,37)(26,27)(28,35)(29,40)(30,36);(9,14,16,15)(10,20,22,21)(11,25,26,34)(12,28,30,29)(13,31,33,32)(17,35,36,40)(18,24,37,27)(19,23,39,38)Polycyclic presentation:
2^9|a^2=g,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,c^b=ch,d^b=df,h^b=hi,g^c=gi,h^c=hi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7410452);