Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996405 |
| 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 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2082 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,7)(6,8)(9,10,14,19,15,20,13,18)(11,21,23,32,25,36,24,16)(12,26,29,34,30,17,28,35)(22,37,27,31,39,33,38,40);(2,5)(3,7)(9,11)(10,16)(12,27)(13,23)(14,24)(15,25)(17,33)(18,21)(19,36)(20,32)(22,29)(26,37)(28,39)(30,38)(31,35)(34,40);(1,2)(3,6)(4,5)(7,8)(9,10)(11,16)(12,17)(13,18)(14,19)(15,20)(21,23)(22,31)(24,36)(25,32)(26,30)(27,33)(28,34)(29,35)(37,38)(39,40);(2,5)(6,8)(9,12,13,28,15,30,14,29)(10,17,18,34,20,26,19,35)(11,22,24,38,25,39,23,27)(16,31,36,37,32,40,21,33);(1,3,4,7)(2,6,5,8)(9,13,15,14)(10,18,20,19)(11,23,25,24)(12,28,30,29)(16,21,32,36)(17,34,26,35)(22,27,39,38)(31,33,40,37)Polycyclic presentation:
2^9|a^2=f,d^2=fi,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,g^c=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7053033);