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 | #60727 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1662 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)(12,26)(13,28)(14,20)(15,32)(16,21)(18,19)(22,37)(23,27)(24,34)(25,36)(29,38)(30,31)(33,35)(39,40);(2,6)(7,8)(9,11,16,25)(10,17,21,36)(12,27,28,38)(13,29,26,23)(14,24,15,33)(18,37,30,40)(19,22,31,39)(20,34,32,35);(1,3,5,4)(2,7,6,8)(9,12,14,31,16,28,15,19)(10,18,32,13,21,30,20,26)(11,22,33,38,25,39,24,27)(17,23,34,40,36,29,35,37);(9,13,14,30,16,26,15,18)(10,19,32,12,21,31,20,28)(11,23,33,40,25,29,24,37)(17,22,34,38,36,39,35,27);(1,4,5,3)(2,7,6,8)(9,14,16,15)(10,20,21,32)(11,24,25,33)(12,19,28,31)(13,30,26,18)(17,34,36,35)(22,38,39,27)(23,37,29,40)Polycyclic presentation:
2^9|b^2=i,c^2=fgi,d^2=gi,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, 7471405);