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 | |
2020 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)(3,5)(4,6)(7,8)(9,10,15,20)(11,21,25,35)(12,17,29,34)(13,18,14,19)(16,23,31,24)(22,37,39,32)(26,40,38,30)(27,33,28,36);(1,2)(3,7)(4,6)(5,8)(9,11)(10,16)(12,26)(13,23)(14,24)(15,25)(17,32)(18,21)(19,35)(20,31)(22,28)(27,39)(29,38)(30,33)(34,37)(36,40);(1,3,4,8)(2,5,6,7)(9,11)(10,16)(12,22)(13,24)(14,23)(15,25)(17,30)(18,35)(19,21)(20,31)(26,28)(27,38)(29,39)(32,33)(34,40)(36,37);(3,8)(5,7)(9,12,15,29)(10,17,20,34)(11,22,25,39)(13,27,14,28)(16,30,31,40)(18,33,19,36)(21,37,35,32)(23,26,24,38);(2,6)(5,7)(9,13,15,14)(10,18,20,19)(11,23,25,24)(12,27,29,28)(16,21,31,35)(17,33,34,36)(22,26,39,38)(30,37,40,32)Polycyclic presentation:
2^9|a^2=i,c^2=h,d^2=i,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=dh,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, 6962130);