Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7563333 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1706 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,4)(3,6,8,7)(9,10)(11,23)(12,16)(13,20)(14,29)(15,21)(17,22)(18,26)(19,31)(24,27)(25,33)(28,35)(30,38)(32,37)(34,40)(36,39);(1,3)(2,6)(4,7)(5,8)(9,11,15,26,17,28,16,27)(10,18,12,23,22,24,21,35)(13,30,31,40,32,39,14,25)(19,36,20,34,29,38,37,33);(1,4,5,2)(3,7,8,6)(9,12,16,10,17,21,15,22)(11,24,26,35,28,18,27,23)(13,20,14,19,32,37,31,29)(25,34,30,38,40,33,39,36);(3,8)(6,7)(9,13,16,14,17,32,15,31)(10,19,21,37,22,29,12,20)(11,25,26,30,28,40,27,39)(18,33,23,36,24,34,35,38);(9,14,15,13,17,31,16,32)(10,20,12,29,22,37,21,19)(11,25,26,30,28,40,27,39)(18,34,23,38,24,33,35,36)Polycyclic presentation:
2^9|a^2=gh,b^2=f,c^2=h,d^2=gi,e^2=f,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7511675);