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 | #61198 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1742 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)(12,25)(13,20)(14,19)(15,18)(16,24)(17,29)(22,38)(23,33)(26,37)(27,36)(28,35)(30,32)(31,40)(34,39);(1,3,5,4)(2,6,7,8)(9,11,14,24)(10,16,19,21)(12,26,28,40)(13,23,15,30)(17,34,36,38)(18,33,20,32)(22,27,39,29)(25,31,35,37);(9,12,14,28)(10,17,19,36)(11,22,24,39)(13,29,15,27)(16,31,21,37)(18,25,20,35)(23,40,30,26)(32,34,33,38);(1,4,5,3)(2,6,7,8)(10,18)(13,15)(16,32)(17,25)(19,20)(21,33)(23,30)(26,40)(27,29)(31,34)(35,36)(37,38);(2,7)(6,8)(9,13,14,15)(10,18,19,20)(11,23,24,30)(12,27,28,29)(16,33,21,32)(17,35,36,25)(22,26,39,40)(31,34,37,38)Polycyclic presentation:
2^9|b^2=f,c^2=i,d^2=fi,e^2=i,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,g^b=gi,e^c=ei,g^c=gi,h^c=hi,e^d=ei,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7424839);