Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667760 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
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 | |
2613 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)(3,7)(4,6)(5,8)(9,10)(11,21)(12,27)(13,20)(14,19)(15,18)(16,25)(17,29)(22,35)(23,24)(26,34)(28,37)(30,32)(31,36)(33,40)(38,39);(1,3,4,8)(2,5,6,7)(9,11,14,25)(10,16,19,21)(12,28,31,40)(13,23,15,26)(17,35,30,39)(18,24,20,34)(22,29,38,32)(27,33,36,37);(9,12)(10,17)(11,22)(13,29)(14,31)(15,32)(16,33)(18,27)(19,30)(20,36)(21,37)(23,28)(24,39)(25,38)(26,40)(34,35);(2,6)(5,7)(9,13,14,15)(10,18,19,20)(11,23,25,26)(12,29,31,32)(16,24,21,34)(17,27,30,36)(22,28,38,40)(33,39,37,35);(3,8)(5,7)(9,10)(11,24)(12,30)(13,18)(14,19)(15,20)(16,26)(17,31)(21,23)(22,35)(25,34)(27,32)(28,33)(29,36)(37,40)(38,39)Polycyclic presentation:
2^9|b^2=f,d^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,e^c=ei,g^c=gi,h^c=hi,h^e=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6575608);