Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514172 |
| 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 | |
2059 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)(11,21)(12,26)(13,28)(14,32)(15,20)(16,25)(17,29)(18,31)(19,34)(22,30)(23,40)(24,37)(27,39)(33,38)(35,36);(1,3,5,4)(2,6,8,7)(9,11,15,25)(10,16,20,21)(12,27,18,36)(13,30,17,23)(14,33,34,24)(19,38,32,37)(22,28,40,29)(26,35,31,39);(9,12)(10,17)(11,22)(13,20)(14,28)(15,18)(16,35)(19,31)(21,39)(23,38)(24,36)(25,40)(26,32)(27,33)(29,34)(30,37);(1,3,5,4)(2,7,8,6)(9,13)(10,18)(11,23)(12,20)(14,26)(15,17)(16,36)(19,29)(21,27)(22,38)(24,35)(25,30)(28,32)(31,34)(33,39)(37,40);(1,4,5,3)(2,7,8,6)(9,14,15,34)(10,19,20,32)(11,24,25,33)(12,28,18,29)(13,26,17,31)(16,37,21,38)(22,36,40,27)(23,35,30,39)Polycyclic presentation:
2^9|b^2=f,d^2=fi,e^2=f,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,e^b=ei,g^b=gi,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, 7429693);