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)(11,16)(12,26)(13,19)(14,20)(15,21)(17,28)(18,23)(22,34)(24,32)(25,33)(27,39)(29,35)(30,36)(31,37)(38,40);(1,3)(2,6)(4,7)(5,8)(9,11)(10,16)(12,27)(13,24)(14,23)(15,25)(17,34)(18,20)(19,32)(21,33)(22,28)(26,39)(29,37)(30,38)(31,35)(36,40);(1,4,5,6)(2,3,7,8)(9,12)(10,17)(11,22)(13,28)(14,30)(15,29)(16,31)(18,34)(19,35)(20,26)(21,36)(23,27)(24,37)(25,38)(32,40)(33,39);(2,7)(3,8)(9,11)(10,18)(12,22)(13,23)(14,24)(15,25)(16,20)(17,34)(19,33)(21,32)(26,31)(27,28)(29,38)(30,37)(35,39)(36,40);(9,13,15,14)(10,19,21,20)(11,23,25,24)(12,28,29,30)(16,18,33,32)(17,35,36,26)(22,27,38,37)(31,34,39,40)Polycyclic presentation:
2^9|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, 7408548);