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 | |
1786 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)(9,10,16,19,15,21,14,20)(11,22,17,28,25,27,24,12)(13,18,31,35,32,37,30,36)(23,34,33,40,38,39,29,26);(1,2)(3,7)(4,6)(5,8)(9,11,14,24,15,25,16,17)(10,12,19,22,21,28,20,27)(13,29,31,23,32,33,30,38)(18,34,36,26,37,39,35,40);(1,3)(2,5)(4,8)(6,7)(9,12)(10,17)(11,19)(13,26)(14,22)(15,28)(16,27)(18,33)(20,25)(21,24)(23,35)(29,37)(30,34)(31,39)(32,40)(36,38);(3,8)(5,7)(9,13)(10,18)(11,23)(12,26)(14,30)(15,32)(16,31)(17,33)(19,35)(20,36)(21,37)(22,34)(24,29)(25,38)(27,39)(28,40);(2,6)(5,7)(9,14,15,16)(10,19,21,20)(11,24,25,17)(12,22,28,27)(13,30,32,31)(18,35,37,36)(23,29,38,33)(26,34,40,39)Polycyclic presentation:
2^9|a^2=gi,b^2=f,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,c^b=ch,d^b=dfi,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, 6910288);