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 | #61177 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1739 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)(11,16)(12,25)(13,27)(14,19)(15,21)(17,24)(18,30)(20,32)(22,38)(23,34)(26,33)(28,39)(29,35)(31,36)(37,40);(2,5)(6,8)(9,11,15,24)(10,16,21,17)(12,14,23,32)(13,28,31,22)(18,33,37,35)(19,34,20,25)(26,40,29,30)(27,39,36,38);(1,3,4,7)(2,6,5,8)(9,12,14,24,15,23,32,11)(10,17,20,25,21,16,19,34)(13,29,30,22,31,26,40,28)(18,33,27,38,37,35,36,39);(9,13,14,30,15,31,32,40)(10,18,20,27,21,37,19,36)(11,22,12,26,24,28,23,29)(16,33,34,38,17,35,25,39);(2,5)(6,8)(9,14,15,32)(10,19,21,20)(11,23,24,12)(13,30,31,40)(16,34,17,25)(18,36,37,27)(22,29,28,26)(33,38,35,39)Polycyclic presentation:
2^9|b^2=i,c^2=fg,d^2=g,e^2=i,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,d^c=di,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, 7467932);