Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519083 |
| 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 | #24231 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1703 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)(4,8)(5,6)(9,10)(11,17)(12,24)(13,26)(14,19)(15,31)(16,20)(18,29)(21,36)(22,33)(23,34)(25,38)(27,39)(28,35)(30,40)(32,37);(2,6)(7,8)(9,11,16,23)(10,17,20,34)(12,14,22,15)(13,27,30,21)(18,32,35,38)(19,33,31,24)(25,29,37,28)(26,39,40,36);(1,3,5,4)(2,7,6,8)(9,12,15,11,16,22,14,23)(10,17,19,24,20,34,31,33)(13,25,29,21,30,37,28,27)(18,32,26,36,35,38,40,39);(1,3,5,4)(2,7,6,8)(9,13,14,28,16,30,15,29)(10,18,31,40,20,35,19,26)(11,21,12,25,23,27,22,37)(17,32,33,39,34,38,24,36);(1,4,5,3)(2,7,6,8)(9,14,16,15)(10,19,20,31)(11,22,23,12)(13,28,30,29)(17,33,34,24)(18,26,35,40)(21,37,27,25)(32,39,38,36)Polycyclic presentation:
2^9|b^2=i,c^2=fg,d^2=fgi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,e^c=ei,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7472633);