Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514160 |
| 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 | #23950 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1720 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)(12,25)(13,30)(14,19)(15,21)(17,28)(18,23)(20,32)(22,34)(24,38)(26,33)(27,35)(29,36)(31,37)(39,40);(1,3,5,4)(2,6,7,8)(9,11,15,13)(10,16,21,30)(12,26,29,39)(14,23,32,24)(17,34,37,35)(18,20,38,19)(22,31,27,28)(25,33,36,40);(1,4,5,3)(2,6,7,8)(9,12)(10,17)(11,22)(13,27)(14,31)(15,29)(16,33)(18,35)(19,25)(20,36)(21,37)(23,26)(24,39)(28,32)(30,40)(34,38);(9,13,15,11)(10,18,21,38)(12,27,29,22)(14,24,32,23)(16,19,30,20)(17,35,37,34)(25,40,36,33)(26,31,39,28);(2,7)(6,8)(9,14,15,32)(10,19,21,20)(11,23,13,24)(12,28,29,31)(16,18,30,38)(17,36,37,25)(22,39,27,26)(33,34,40,35)Polycyclic presentation:
2^9|b^2=fi,c^2=f,d^2=i,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,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7422653);