Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519063 |
| 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 | #3058 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1658 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10)(11,22)(12,17)(13,18)(14,29)(15,21)(16,23)(19,26)(20,27)(24,38)(25,32)(28,31)(30,33)(34,39)(35,40)(36,37);(1,3,5,2)(4,6,8,7)(9,11,12,23,15,13,27,25)(10,16,20,22,21,32,17,18)(14,24,26,37,30,28,40,39)(19,31,29,36,35,38,33,34);(1,4,5,8)(2,6,3,7)(9,12,15,27)(10,17,21,20)(11,23,13,25)(14,26,30,40)(16,18,32,22)(19,33,35,29)(24,37,28,39)(31,34,38,36);(2,3)(6,7)(9,13)(10,18)(11,15)(12,25)(14,28)(16,20)(17,32)(19,34)(21,22)(23,27)(24,30)(26,39)(29,31)(33,38)(35,36)(37,40);(9,14,27,40,15,30,12,26)(10,19,17,33,21,35,20,29)(11,24,25,39,13,28,23,37)(16,31,18,34,32,38,22,36)Polycyclic presentation:
2^9|a^2=g,b^2=fg,c^2=gi,e^2=fi,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,h^b=hi,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7453124);