Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806827 |
| 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 |
| prime radical | #60765 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1943 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)(11,22)(12,27)(13,20)(14,19)(15,31)(16,21)(17,25)(18,24)(23,40)(26,35)(28,38)(29,39)(30,37)(32,34)(33,36);(1,3)(2,5)(4,8)(6,7)(9,11)(10,17)(12,13)(14,25)(15,26)(16,33)(18,37)(19,22)(20,38)(21,35)(23,24)(27,39)(28,29)(30,32)(31,36)(34,40);(9,12,16,24,14,28,15,32)(10,18,21,38,19,34,31,27)(11,23,26,13,25,30,33,29)(17,20,36,37,22,39,35,40);(2,6)(5,7)(10,19)(17,22)(18,34)(20,39)(21,31)(27,38)(35,36)(37,40);(3,8)(5,7)(9,13,16,30,14,29,15,23)(10,20,21,37,19,39,31,40)(11,24,26,28,25,32,33,12)(17,34,36,27,22,18,35,38)Polycyclic presentation:
2^9|c^2=h,e^2=h,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=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6714023);