Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514148 |
| 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 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1738 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,8)(4,6)(5,7)(9,10)(11,20)(12,25)(13,30)(14,19)(15,23)(16,29)(17,24)(18,31)(21,35)(22,33)(26,38)(27,37)(28,36)(32,40)(34,39);(1,3,5,4)(2,6,7,8)(9,11)(10,15)(12,26)(13,22)(14,23)(16,34)(17,30)(18,33)(19,20)(21,29)(24,31)(25,32)(27,39)(28,40)(35,37)(36,38);(1,4,5,3)(2,6,7,8)(9,12,14,28)(10,16,19,37)(11,21,23,39)(13,27,31,29)(15,32,20,38)(17,35,33,34)(18,36,30,25)(22,26,24,40);(9,11)(10,17)(12,21)(13,22)(14,23)(15,18)(16,35)(19,33)(20,30)(24,31)(25,38)(26,27)(28,39)(29,40)(32,36)(34,37);(2,7)(6,8)(9,13,14,31)(10,18,19,30)(11,22,23,24)(12,27,28,29)(15,33,20,17)(16,36,37,25)(21,26,39,40)(32,34,38,35)Polycyclic presentation:
2^9|b^2=fi,c^2=f,e^2=i,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,g^b=gi,g^c=gi,h^c=hi,g^d=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7422639);