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 |
| prime radical | #23881 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1908 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)(3,7)(4,6)(5,8)(9,10)(11,12)(13,20)(14,21)(15,22)(16,36)(17,23)(18,26)(19,30)(24,28)(25,29)(27,31)(32,37)(33,38)(34,39)(35,40);(1,3,5,4)(2,6,8,7)(9,11)(10,18)(12,22)(13,24)(14,20)(15,26)(16,27)(17,19)(21,33)(23,31)(25,35)(28,38)(29,37)(30,36)(32,39)(34,40);(1,3,5,4)(2,7,8,6)(9,12,15,18)(10,19,22,27)(11,23,26,36)(13,32,33,35)(14,34,28,29)(16,31,17,30)(20,21,38,24)(25,40,39,37);(1,4,5,3)(2,6,8,7)(9,13)(10,20)(11,24)(12,28)(14,18)(15,33)(16,34)(17,29)(19,37)(21,26)(22,38)(23,25)(27,40)(30,32)(31,35)(36,39);(9,14,15,28)(10,21,22,24)(11,25,26,39)(12,29,18,34)(13,31,33,30)(16,35,17,32)(19,20,27,38)(23,37,36,40)Polycyclic presentation:
2^9|b^2=fi,c^2=f,d^2=fi,e^2=i,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, 6484597);