Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514148 |
| 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 | #61197 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1748 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)(3,8)(4,6)(5,7)(9,10)(11,16)(12,24)(13,29)(14,18)(15,20)(17,27)(19,30)(21,33)(22,35)(23,32)(25,31)(26,34)(28,36)(37,39)(38,40);(1,3,5,4)(2,6,7,8)(9,11)(10,16)(12,25)(13,14)(15,23)(17,33)(18,29)(19,35)(20,32)(21,27)(22,30)(24,31)(26,37)(28,38)(34,39)(36,40);(9,12,15,28)(10,17,20,34)(11,21,23,37)(13,25,22,38)(14,26,30,27)(16,31,32,40)(18,24,19,36)(29,39,35,33);(1,4,5,3)(2,6,7,8)(9,13,15,22)(10,16,20,32)(11,14,23,30)(12,25,28,38)(17,31,34,40)(18,29,19,35)(21,26,37,27)(24,39,36,33);(2,7)(6,8)(9,14,15,30)(10,18,20,19)(11,13,23,22)(12,26,28,27)(16,29,32,35)(17,24,34,36)(21,25,37,38)(31,39,40,33)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,g^c=gi,h^c=hi,g^d=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7424791);