Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514172 |
| 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 |
| prime radical | C2×2−1+4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1648 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,8)(3,7,4,6)(9,10,15,19)(11,16,22,35)(12,24,27,28)(13,26,17,31)(14,29,32,18)(20,37,39,21)(23,38,30,36)(25,33,34,40);(1,3,5,4)(2,6,8,7)(9,11)(10,16)(12,25)(13,21)(14,30)(15,22)(17,37)(18,38)(19,35)(20,26)(23,32)(24,33)(27,34)(28,40)(29,36)(31,39);(9,12)(10,17)(11,20)(13,19)(14,31)(15,27)(16,33)(18,28)(21,36)(22,39)(23,34)(24,29)(25,30)(26,32)(35,40)(37,38);(9,13,15,17)(10,12,19,27)(11,21,22,37)(14,24,32,28)(16,34,35,25)(18,31,29,26)(20,36,39,38)(23,40,30,33);(1,4,5,3)(2,7,8,6)(11,22)(16,35)(20,39)(21,37)(23,30)(25,34)(33,40)(36,38)Polycyclic presentation:
2^9|a^2=fi,b^2=f,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,e^b=ei,g^b=gi,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, 7426512);