Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574084 |
| 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 | |
1977 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)(9,10,14,11)(12,17,26,22)(13,18,31,23)(15,20,24,19)(16,28,21,27)(25,35,39,38)(29,33,36,32)(30,34,40,37);(1,3,5,4)(2,6,7,8)(9,11)(10,14)(12,22)(13,29)(15,20)(16,27)(17,26)(18,32)(19,24)(21,28)(23,33)(25,37)(30,38)(31,36)(34,39)(35,40);(9,12,24,28,14,26,15,27)(10,16,20,22,11,21,19,17)(13,25,33,40,31,39,32,30)(18,34,29,38,23,37,36,35);(1,4,5,3)(2,6,7,8)(9,12,14,26)(10,17,11,22)(13,30,31,40)(15,27,24,28)(16,19,21,20)(18,34,23,37)(25,32,39,33)(29,38,36,35);(9,13)(10,18)(11,23)(12,25)(14,31)(15,32)(16,34)(17,35)(19,36)(20,29)(21,37)(22,38)(24,33)(26,39)(27,30)(28,40)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=gi,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,h^b=hi,d^c=dh,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, 6491515);