Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993875 |
| 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 | |
1791 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)(9,10,15,30,16,20,14,19)(11,21,24,13,18,32,23,31)(12,25,27,17,29,34,28,35)(22,26,37,36,33,40,38,39);(1,2)(3,7)(4,6)(5,8)(9,11,14,23,16,18,15,24)(10,13,30,32,20,31,19,21)(12,22,27,37,29,33,28,38)(17,26,25,39,35,40,34,36);(1,3)(2,5)(4,8)(6,7)(9,12)(10,17)(11,22)(13,26)(14,27)(15,28)(16,29)(18,33)(19,34)(20,35)(21,36)(23,37)(24,38)(25,30)(31,40)(32,39);(3,8)(5,7)(9,13)(10,18)(11,20)(12,26)(14,21)(15,32)(16,31)(17,33)(19,24)(22,35)(23,30)(25,37)(27,36)(28,39)(29,40)(34,38);(2,6)(5,7)(9,14,16,15)(10,19,20,30)(11,23,18,24)(12,27,29,28)(13,21,31,32)(17,34,35,25)(22,37,33,38)(26,36,40,39)Polycyclic presentation:
2^9|a^2=gi,b^2=fi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,d^c=dh,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, 6910259);