Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519075 |
| 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 | #24234 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1723 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,7)(4,8)(5,6)(9,10)(11,20)(12,23)(13,27)(14,18)(15,30)(16,22)(17,29)(19,24)(21,35)(25,32)(26,34)(28,40)(31,39)(33,38)(36,37);(2,6)(7,8)(9,11,14,22)(10,16,18,20)(12,15,26,24)(13,21,28,36)(17,32,33,31)(19,34,30,23)(25,29,39,38)(27,37,40,35);(1,3,5,4)(2,7,6,8)(9,12,24,11,14,26,15,22)(10,16,30,34,18,20,19,23)(13,25,38,21,28,39,29,36)(17,31,40,35,33,32,27,37);(1,3,5,4)(2,7,6,8)(9,13,24,38,14,28,15,29)(10,17,30,40,18,33,19,27)(11,21,26,39,22,36,12,25)(16,31,34,35,20,32,23,37);(1,4,5,3)(2,7,6,8)(10,18)(16,20)(17,33)(19,30)(23,34)(27,40)(31,32)(35,37)Polycyclic presentation:
2^9|b^2=i,c^2=fg,d^2=fg,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=cgi,d^b=dh,g^b=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7472267);