Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #232789 |
| 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 |
| prime radical | #60940 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1398 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)(15,22)(17,27)(20,35)(21,33)(23,31)(24,39)(26,29)(28,40)(30,36)(32,38)(34,37);(1,2,4,6)(3,8,7,5)(9,11,14,21)(10,15,18,29)(12,20,23,37)(13,26,27,22)(16,30,31,28)(17,33,25,19)(24,40,38,36)(32,34,39,35);(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);(11,21)(12,23)(15,29)(16,31)(19,33)(22,26)(24,38)(32,39)Polycyclic presentation:
2^9|b^2=h,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=dhi,e^b=eh,f^b=fh,d^c=dhi,e^c=ei,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72303255, #72303308, #72303381, … |
| maximal normal subgroup of | #72303255, #72303308, #72303381, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 140817);