Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470037 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3992 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)(4,7)(5,8)(10,16)(13,14)(17,32)(18,31)(19,28)(20,30)(22,29)(24,34)(25,27)(33,40)(35,39)(36,37);(3,4)(6,7)(9,10)(11,20)(12,24)(13,16)(14,28)(15,19)(17,23)(18,26)(21,33)(22,32)(25,31)(27,34)(29,30)(35,38)(36,40)(37,39);(1,3,5,4)(2,6,8,7)(9,11)(10,17)(12,21)(13,22)(14,29)(15,23)(16,30)(18,33)(19,20)(24,35)(25,36)(26,38)(27,37)(28,32)(31,39)(34,40);(1,4,5,3)(2,7,8,6)(9,12,15,26)(10,18,19,24)(11,21,23,38)(13,27,14,25)(16,31,28,34)(17,33,20,35)(22,37,29,36)(30,39,32,40);(2,8)(6,7)(9,13,15,14)(10,16,19,28)(11,22,23,29)(12,25,26,27)(17,30,20,32)(18,34,24,31)(21,36,38,37)(33,40,35,39)Polycyclic presentation:
2^9|c^2=hi,d^2=h,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,g^b=gi,f^c=fi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6394705);