Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574077 |
| 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 | |
1981 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)(3,8)(4,6)(5,7)(9,10)(11,21)(12,17)(13,18)(14,20)(15,25)(16,30)(19,32)(22,36)(23,33)(24,27)(26,35)(28,34)(29,38)(31,37)(39,40);(1,3,5,4)(2,6,7,8)(9,11,13,25)(10,15,18,21)(12,28,29,23)(14,26,32,27)(16,36,37,40)(17,33,38,34)(19,35,20,24)(22,30,39,31);(9,12,14,30,13,29,32,31)(10,16,19,17,18,37,20,38)(11,22,26,28,25,39,27,23)(15,33,35,40,21,34,24,36);(1,4,5,3)(2,6,7,8)(9,12,13,29)(10,17,18,38)(11,23,25,28)(14,30,32,31)(15,34,21,33)(16,19,37,20)(22,27,39,26)(24,40,35,36);(9,10)(11,24)(12,16)(13,18)(14,19)(15,26)(17,30)(20,32)(21,27)(22,36)(23,34)(25,35)(28,33)(29,37)(31,38)(39,40)Polycyclic presentation:
2^9|b^2=f,c^2=g,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6491528);