Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #232771 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
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 | |
1859 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,6)(4,7)(5,8)(9,10)(11,17)(12,20)(13,16)(14,19)(15,22)(18,24)(21,23);(2,6)(3,4)(7,8)(9,11)(10,14)(12,21)(13,18)(15,24)(16,23)(17,20)(19,22);(2,7)(3,4)(6,8)(14,23)(17,24);(11,18)(14,23)(17,24)(19,21)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=dhi,e^b=eh,f^b=fh,e^c=ei,f^c=fi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72239035, #72240781, #72257360, … |
| maximal normal subgroup of | #72239035, #72240781, #72257360, … |
| maximal quotient of | #72237257, #72239702 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 140784);