Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500740 |
| 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 | |
1758 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)(11,21)(12,27)(13,29)(14,19)(15,20)(16,34)(17,25)(18,32)(22,31)(23,40)(24,37)(26,36)(28,39)(30,33)(35,38);(1,3)(2,6)(4,7)(5,8)(9,11)(10,17)(12,28)(13,31)(14,24)(15,25)(16,26)(18,38)(19,36)(20,21)(22,30)(23,33)(27,35)(29,40)(32,39)(34,37);(9,12,15,18)(10,13,20,33)(11,22,25,40)(14,29,16,30)(17,35,21,39)(19,32,34,27)(23,36,31,37)(24,28,26,38);(2,7)(3,8)(9,13)(10,18)(11,23)(12,20)(14,27)(15,33)(16,32)(17,28)(19,29)(21,38)(22,36)(24,39)(25,31)(26,35)(30,34)(37,40);(1,4,5,6)(2,3,7,8)(9,14,15,16)(10,19,20,34)(11,24,25,26)(12,29,18,30)(13,32,33,27)(17,36,21,37)(22,28,40,38)(23,35,31,39)Polycyclic presentation:
2^9|c^2=i,e^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,h^b=hi,g^c=gi,h^c=hi,e^d=ei,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7409739);