Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993887 |
| 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 | |
2084 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)(9,10,15,20)(11,21,24,37)(12,17,28,36)(13,19,14,18)(16,29,32,23)(22,39,26,33)(25,34,40,31)(27,38,30,35);(1,2)(3,7)(4,6)(5,8)(9,11)(10,16)(12,25)(13,29)(14,23)(15,24)(17,33)(18,37)(19,21)(20,32)(22,27)(26,30)(28,40)(31,35)(34,38)(36,39);(1,3,4,8)(2,5,6,7)(9,11)(10,16)(12,26)(13,23)(14,29)(15,24)(17,34)(18,21)(19,37)(20,32)(22,28)(25,30)(27,40)(31,36)(33,38)(35,39);(3,8)(5,7)(9,12)(10,17)(11,22)(13,30)(14,27)(15,28)(16,31)(18,38)(19,35)(20,36)(21,39)(23,40)(24,26)(25,29)(32,34)(33,37);(2,6)(5,7)Polycyclic presentation:
2^9|a^2=i,c^2=h,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,g^c=gi,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6962143);