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 | |
1782 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)(9,10,13,18,14,19,31,17)(11,20,23,21,25,33,24,15)(12,16,28,36,30,27,29,35)(22,34,38,37,39,40,26,32);(1,2)(3,7)(4,6)(5,8)(9,11)(10,15)(12,26)(13,23)(14,25)(16,34)(17,33)(18,20)(19,21)(22,28)(24,31)(27,40)(29,39)(30,38)(32,35)(36,37);(1,3,4,8)(2,5,6,7)(9,10)(11,21)(12,27)(13,17)(14,19)(15,25)(16,30)(18,31)(20,23)(22,32)(24,33)(26,34)(28,36)(29,35)(37,39)(38,40);(3,8)(5,7)(9,12)(10,16)(11,22)(13,28)(14,30)(15,32)(17,35)(18,36)(19,27)(20,34)(21,37)(23,38)(24,26)(25,39)(29,31)(33,40);(2,6)(5,7)(9,13,14,31)(10,17,19,18)(11,23,25,24)(12,28,30,29)(15,33,21,20)(16,35,27,36)(22,38,39,26)(32,40,37,34)Polycyclic presentation:
2^9|a^2=g,c^2=hi,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=dfi,e^b=ehi,f^b=fi,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, 6970336);