Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514160 |
| 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 | |
1740 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)(11,19)(12,24)(13,15)(14,28)(16,23)(17,26)(18,27)(20,30)(21,32)(22,33)(25,34)(29,31)(35,36)(37,40)(38,39);(2,6)(7,8)(10,15)(11,20)(14,27)(16,19)(21,37)(22,23)(24,34)(26,29)(30,33)(32,35)(36,38)(39,40);(1,3,5,4)(2,7,6,8)(9,11,13,22)(10,16,15,30)(12,21,25,38)(14,20,27,23)(17,32,31,39)(18,33,28,19)(24,35,34,40)(26,37,29,36);(9,12,13,25)(10,17,15,31)(11,21,22,38)(14,29,27,26)(16,32,30,39)(18,34,28,24)(19,35,33,40)(20,36,23,37);(1,4,5,3)(2,7,6,8)(10,15)(16,30)(17,31)(18,28)(19,33)(24,34)(32,39)(35,40)Polycyclic presentation:
2^9|c^2=f,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cgi,d^b=dh,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, 7426921);