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 |
| prime radical | #60450 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2122 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)(9,10,15,20)(11,16,23,30)(12,25,27,33)(13,17,14,19)(18,28,34,26)(21,31,24,32)(22,36,37,40)(29,38,39,35);(1,3)(2,6)(4,7)(5,8)(9,11)(10,16)(12,22)(13,24)(14,21)(15,23)(17,32)(18,29)(19,31)(20,30)(25,36)(26,35)(27,37)(28,38)(33,40)(34,39);(1,4,5,6)(2,3,7,8)(10,17)(11,21)(13,14)(18,25)(19,20)(22,35)(23,24)(26,28)(31,32)(33,34)(36,40)(37,38);(2,7)(3,8)(9,12)(10,18)(11,22)(13,28)(14,26)(15,27)(16,29)(17,25)(19,33)(20,34)(21,35)(23,37)(24,38)(30,39)(31,40)(32,36);(11,23)(16,30)(21,24)(22,37)(29,39)(31,32)(35,38)(36,40)Polycyclic presentation:
2^9|a^2=i,c^2=f,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, 7408547);