Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500908 |
| 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 | |
1791 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)(9,10,13,19,14,18,28,16)(11,15,20,30,22,32,23,31)(12,24,25,17,27,33,26,34)(21,36,38,29,37,39,35,40);(1,3)(2,6)(4,7)(5,8)(9,11)(10,15)(12,21)(13,20)(14,22)(16,31)(17,29)(18,32)(19,30)(23,28)(24,36)(25,38)(26,35)(27,37)(33,39)(34,40);(10,16)(11,20)(13,28)(17,33)(18,19)(21,35)(22,23)(24,34)(25,26)(30,31)(36,39)(37,38);(2,7)(3,8)(9,12)(10,17)(11,21)(13,26)(14,27)(15,29)(16,33)(18,34)(19,24)(20,35)(22,37)(23,38)(25,28)(30,36)(31,39)(32,40);(1,4,5,6)(2,3,7,8)(10,18)(11,22)(16,19)(17,34)(20,23)(21,37)(24,33)(35,38)Polycyclic presentation:
2^9|a^2=g,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,e^a=ei,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, 7411150);