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 | |
1764 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)(11,23)(12,29)(13,32)(14,22)(15,21)(16,20)(17,27)(18,31)(19,33)(24,25)(26,35)(28,36)(30,39)(34,38)(37,40);(1,3,5,4)(2,6,8,7)(9,11,15,27)(10,17,21,23)(12,30,19,38)(13,25,18,37)(14,28,16,26)(20,36,22,35)(24,32,40,31)(29,34,33,39);(1,3,5,4)(2,7,8,6)(9,12,15,19)(10,18,21,13)(11,24,27,40)(14,31,16,32)(17,34,23,39)(20,29,22,33)(25,35,37,36)(26,30,28,38);(9,13,15,18)(10,19,21,12)(11,25,27,37)(14,33,16,29)(17,30,23,38)(20,32,22,31)(24,35,40,36)(26,34,28,39);(1,4,5,3)(2,7,8,6)(9,14,15,16)(10,20,21,22)(11,26,27,28)(12,31,19,32)(13,33,18,29)(17,35,23,36)(24,30,40,38)(25,34,37,39)Polycyclic presentation:
2^9|b^2=f,c^2=f,d^2=i,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, 7427443);