Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437478 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3920 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)(11,21)(12,17)(13,18)(14,19)(15,20)(16,25)(22,26)(23,37)(24,32)(27,31)(28,34)(29,36)(30,35)(33,39)(38,40);(9,11,15,26)(10,16,20,32)(12,27,29,38)(13,24,14,25)(17,33,36,37)(18,21,19,22)(23,30,39,28)(31,35,40,34);(1,3,5,4)(2,6,7,8)(9,10)(11,22)(12,17)(13,19)(14,18)(15,20)(16,25)(21,26)(23,33)(24,32)(27,31)(28,35)(29,36)(30,34)(37,39)(38,40);(1,4,5,3)(2,6,7,8)(9,12)(10,17)(11,23)(13,28)(14,30)(15,29)(16,31)(18,34)(19,35)(20,36)(21,37)(22,33)(24,38)(25,27)(26,39)(32,40);(9,13,15,14)(10,18,20,19)(11,24,26,25)(12,28,29,30)(16,21,32,22)(17,34,36,35)(23,38,39,27)(31,37,40,33)Polycyclic presentation:
2^9|b^2=i,c^2=g,d^2=g,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72318726, #72318970, #72319796, … |
| maximal normal subgroup of | #72318726, #72318970, #72319796, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6358967);