Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514172 |
| 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 | |
1786 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,8)(4,6)(5,7)(9,10)(11,16)(12,24)(13,19)(14,18)(15,20)(17,27)(21,34)(22,31)(23,33)(25,30)(26,35)(28,36)(29,32)(37,39)(38,40);(1,3,5,4)(2,6,7,8)(9,11,15,23)(10,16,20,33)(12,25,28,38)(13,22,14,29)(17,34,35,39)(18,32,19,31)(21,26,37,27)(24,30,36,40);(9,12,15,28)(10,17,20,35)(11,21,23,37)(13,27,14,26)(16,30,33,40)(18,24,19,36)(22,38,29,25)(31,34,32,39);(1,4,5,3)(2,6,7,8)(10,18)(13,14)(16,31)(17,24)(19,20)(22,29)(25,38)(26,27)(30,34)(32,33)(35,36)(39,40);(2,7)(6,8)(9,13,15,14)(10,19,20,18)(11,22,23,29)(12,26,28,27)(16,31,33,32)(17,24,35,36)(21,25,37,38)(30,39,40,34)Polycyclic presentation:
2^9|b^2=fi,c^2=i,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,e^c=ei,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, 7424845);