Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573643 |
| 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 | |
2612 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)(11,18)(12,26)(13,20)(14,21)(15,32)(16,22)(17,23)(19,28)(24,35)(25,36)(27,34)(29,38)(30,39)(31,37)(33,40);(1,3,5,4)(2,6,7,8)(9,11)(10,18)(12,27)(13,16)(14,29)(15,24)(17,25)(19,37)(20,22)(21,38)(23,36)(26,34)(28,31)(30,33)(32,35)(39,40);(9,12,17,30)(10,19,23,38)(11,14,25,31)(13,27,24,33)(15,29,16,28)(18,34,36,40)(20,37,35,21)(22,39,32,26);(1,4,5,3)(2,6,7,8)(9,13,17,24)(10,20,23,35)(11,16,25,15)(12,14,30,31)(18,22,36,32)(19,34,38,40)(21,39,37,26)(27,29,33,28);(9,14)(10,21)(11,12)(13,29)(15,27)(16,33)(17,31)(18,26)(19,35)(20,38)(22,40)(23,37)(24,28)(25,30)(32,34)(36,39)Polycyclic presentation:
2^9|b^2=f,c^2=i,d^2=fi,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,e^c=ei,g^c=gi,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6483942);