Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500736 |
| 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 | |
2140 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)(3,4)(5,7)(6,8)(9,10,15,21)(11,16,25,33)(12,26,30,37)(13,19,14,20)(17,28,36,29)(18,23,32,24)(22,35,38,34)(27,31,39,40);(1,3)(2,6)(4,7)(5,8)(9,11)(10,16)(12,27)(13,24)(14,23)(15,25)(17,34)(18,19)(20,32)(21,33)(22,28)(26,31)(29,38)(30,39)(35,36)(37,40);(1,4,5,6)(2,3,7,8)(9,12)(10,17)(11,22)(13,28)(14,29)(15,30)(16,31)(18,35)(19,26)(20,37)(21,36)(23,27)(24,39)(25,38)(32,34)(33,40);(2,7)(3,8)(9,11)(10,18)(12,22)(13,23)(14,24)(15,25)(16,19)(17,35)(20,33)(21,32)(26,31)(27,28)(29,39)(30,38)(34,36)(37,40);(9,13,15,14)(10,19,21,20)(11,23,25,24)(12,28,30,29)(16,32,33,18)(17,26,36,37)(22,27,38,39)(31,34,40,35)Polycyclic presentation:
2^9|a^2=i,c^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,g^a=gi,c^b=ch,d^b=df,e^b=ei,h^b=hi,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, 7408549);