Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667760 |
| 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 | |
1999 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,23)(13,21)(14,19)(16,18)(17,28)(20,25)(22,32)(24,27)(26,37)(29,30)(31,38)(33,35)(34,40)(36,39);(2,5)(3,7)(9,11,15,10)(12,24,27,23)(13,21,16,18)(14,30,32,35)(17,25,20,28)(19,33,22,29)(26,34,31,36)(37,39,38,40);(1,3,4,7)(2,6,5,8)(9,12,15,27)(10,17,11,20)(13,25,16,28)(14,31,32,26)(18,24,21,23)(19,36,22,34)(29,40,33,39)(30,38,35,37);(2,5)(6,8)(9,13,15,16)(10,18,11,21)(12,25,27,28)(14,29,32,33)(17,24,20,23)(19,35,22,30)(26,39,31,40)(34,38,36,37);(9,14,15,32)(10,19,11,22)(12,26,27,31)(13,29,16,33)(17,34,20,36)(18,35,21,30)(23,37,24,38)(25,39,28,40)Polycyclic presentation:
2^9|b^2=i,c^2=f,d^2=i,e^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, 6576263);