Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10496 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #394 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C22 | #1440 |
| prime radical | #3444 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
578 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,5,8,7,3,6,4)(9,10)(11,21,25,16)(12,26)(13,20)(14,19)(15,18)(17,28)(22,38,39,34)(23,32,24,33)(27,37,40,31)(29,36)(30,35);(1,3)(2,7)(4,5)(6,8)(9,11)(10,16)(12,27)(13,24)(14,23)(15,25)(17,34)(18,21)(19,33)(20,32)(22,28)(26,31)(29,40)(30,39)(35,38)(36,37);(1,4,5,3,7,8,6,2)(9,12)(10,17,18,35)(11,22)(13,28)(14,30)(15,29)(16,31,21,37)(19,26,20,36)(23,27)(24,40)(25,39)(32,34,33,38)Polycyclic presentation:
2^9|a^2=g,c^2=d,d^2=i,e^2=hi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,c^b=cf,d^b=di,e^b=eh,f^b=fh,g^b=ghi,e^c=ei,f^c=fhi,g^c=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 26291);