Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #232809 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#127 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#392 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1346 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,5)(3,7)(6,8)(9,10)(11,19)(12,16)(13,25)(14,18)(15,21)(17,27)(20,35)(22,33)(23,31)(24,39)(26,29)(28,40)(30,37)(32,38)(34,36);(1,2)(3,8)(4,6)(5,7)(9,11)(10,15)(12,20)(13,26)(14,22)(16,30)(17,33)(18,29)(19,25)(21,27)(23,36)(24,40)(28,31)(32,34)(35,39)(37,38);(1,2)(3,8)(4,6)(5,7)(9,12)(10,16)(11,20)(13,24)(14,23)(15,28)(17,32)(18,31)(19,34)(21,37)(22,36)(25,39)(26,40)(27,38)(29,30)(33,35);(2,6)(5,8)(12,23)(16,31)(20,36)(24,38)(28,30)(32,39)(34,35)(37,40)Polycyclic presentation:
2^9|e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,d^b=di,e^b=eh,f^b=fh,d^c=dh,e^c=ei,f^c=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72303175, #72303184, #72303464, … |
| maximal normal subgroup of | #72303175, #72303184, #72303464, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 140793);