Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514160 |
| 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 | |
1760 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)(11,22)(12,25)(13,31)(14,21)(15,20)(16,19)(17,24)(18,30)(23,39)(26,38)(27,33)(28,36)(29,35)(32,37)(34,40);(1,3,5,4)(2,6,7,8)(9,11)(10,17)(12,26)(13,14)(15,24)(16,32)(18,34)(19,31)(20,22)(21,37)(23,30)(25,33)(27,29)(28,40)(35,38)(36,39);(9,12)(10,18)(11,23)(13,27)(14,30)(15,29)(16,28)(17,33)(19,25)(20,36)(21,35)(22,38)(24,40)(26,32)(31,39)(34,37);(1,4,5,3)(2,6,7,8)(9,13,15,32)(10,17,20,22)(11,14,24,16)(12,27,29,26)(18,33,36,38)(19,31,21,37)(23,30,40,28)(25,39,35,34);(2,7)(6,8)(9,14,15,16)(10,19,20,21)(11,13,24,32)(12,28,29,30)(17,31,22,37)(18,35,36,25)(23,26,40,27)(33,34,38,39)Polycyclic presentation:
2^9|b^2=fi,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,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7424825);