Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C22×C22⋊C4 |
| prime radical | #34964 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1044 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,20)(12,19)(13,18)(14,17)(15,22)(16,21)(23,24)(25,26,29,33,30,34,28,32)(27,35,37,31,36,40,38,39);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,23)(13,22)(14,21)(16,19)(17,24)(18,20)(27,36)(31,39)(35,40)(37,38);(3,8)(5,7)(9,11)(10,16)(12,21)(13,22)(14,23)(15,19)(17,20)(18,24)(25,27)(26,31)(28,38)(29,37)(30,36)(32,35)(33,40)(34,39);(2,6)(5,7)(9,12,13,14)(10,17,18,19)(11,21,22,23)(15,16,20,24)(25,28,30,29)(26,32,34,33)(27,37,36,38)(31,40,39,35)Polycyclic presentation:
2^9|a^2=g,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ceh,d^b=dhi,d^c=di,f^c=fh80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 121468);