Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993899 |
| 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 | |
2045 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)(3,5)(4,6)(7,8)(9,10)(11,21)(12,17)(13,18)(14,19)(15,20)(16,24)(22,39)(23,32)(25,37)(26,34)(27,33)(28,35)(29,38)(30,36)(31,40);(1,2)(3,7)(4,6)(5,8)(9,11)(10,16)(12,26)(13,23)(14,24)(15,25)(17,33)(18,21)(19,37)(20,32)(22,29)(27,28)(30,40)(31,35)(34,38)(36,39);(1,3,4,8)(2,5,6,7)(9,11)(10,16)(12,27)(13,24)(14,23)(15,25)(17,34)(18,37)(19,21)(20,32)(22,30)(26,28)(29,40)(31,36)(33,38)(35,39);(3,8)(5,7)(9,12)(10,17)(11,22)(13,28)(14,29)(15,30)(16,31)(18,35)(19,38)(20,36)(21,39)(23,26)(24,40)(25,27)(32,34)(33,37);(2,6)(5,7)(9,13,15,14)(10,18,20,19)(11,23,25,24)(12,28,30,29)(16,21,32,37)(17,35,36,38)(22,26,27,40)(31,39,34,33)Polycyclic presentation:
2^9|c^2=h,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,e^c=ei,g^c=gi,f^d=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6962152);