Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573635 |
| 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 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1889 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)(4,8)(5,6)(9,10)(11,23)(12,19)(13,20)(14,21)(15,18)(16,22)(17,26)(24,33)(25,36)(27,35)(28,40)(29,37)(30,34)(31,38)(32,39);(2,6)(7,8)(9,11)(10,17)(12,15)(13,28)(14,26)(16,24)(18,33)(19,22)(20,35)(21,23)(25,30)(27,29)(31,39)(32,34)(36,38)(37,40);(10,18)(11,24)(12,26)(15,16)(19,33)(20,34)(21,22)(25,28)(27,39)(30,31)(35,40)(37,38);(1,3,5,4)(2,7,6,8)(9,12)(10,19)(11,16)(13,25)(14,24)(15,26)(17,18)(20,36)(21,33)(22,23)(27,30)(28,31)(29,39)(32,37)(34,35)(38,40);(1,4,5,3)(2,8,6,7)(9,13)(10,20)(11,25)(12,27)(14,29)(15,30)(16,31)(17,32)(18,34)(19,35)(21,37)(22,38)(23,36)(24,28)(26,39)(33,40)Polycyclic presentation:
2^9|d^2=fi,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,h^b=hi,d^c=dh,g^c=gi,h^c=hi,e^d=eh,h^d=hi,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6485236);