Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667964 |
| 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 |
| prime radical | #60376 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2571 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)(3,7)(4,6)(5,8)(9,10)(11,16)(12,26)(13,18)(14,19)(15,20)(17,28)(21,34)(22,32)(23,24)(25,33)(27,39)(29,35)(30,36)(31,37)(38,40);(1,3,4,8)(2,5,6,7)(9,11,15,25)(10,16,20,33)(12,27,29,37)(13,22,14,24)(17,34,36,40)(18,32,19,23)(21,30,38,28)(26,39,35,31);(9,12)(10,17)(11,21)(13,28)(14,30)(15,29)(16,31)(18,35)(19,26)(20,36)(22,37)(23,34)(24,27)(25,38)(32,40)(33,39);(2,6)(5,7)(9,13,15,14)(10,18,20,19)(11,22,25,24)(12,28,29,30)(16,32,33,23)(17,35,36,26)(21,37,38,27)(31,40,39,34);(3,8)(5,7)(9,10)(11,23)(12,17)(13,19)(14,18)(15,20)(16,24)(21,34)(22,33)(25,32)(26,28)(27,31)(29,36)(30,35)(37,39)(38,40)Polycyclic presentation:
2^9|b^2=fi,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,g^c=gi,h^c=hi,e^d=ei,h^e=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6575818);