Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #235438 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#392 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1623 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,10)(4,8)(5,9)(6,14)(7,15)(11,18)(12,17)(13,22)(16,24)(19,30)(20,26)(21,28)(23,29)(25,32)(27,31);(2,8)(4,18)(5,21)(9,27)(10,12)(11,14)(13,25)(15,26)(17,23)(20,32)(24,29)(28,30);(2,9)(3,16)(4,20)(5,17)(8,10)(11,24)(12,13)(14,28)(15,29)(18,32)(19,31)(21,23)(25,27)(26,30);(1,3,7,19)(2,10,15,30)(4,17,20,21)(5,18,23,32)(6,16,22,31)(8,12,26,28)(9,11,29,25)(13,27,14,24)Polycyclic presentation:
2^9|d^2=i,e^2=h,f^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fhi,g^a=ghi,c^b=cg,e^b=eh,f^b=fh,d^c=dh,e^c=ei,f^c=fhi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72213685, #72213763, #72213848, … |
| maximal normal subgroup of | #72213685, #72213763, #72213848, … |
| maximal quotient of | C2×#235444, #72240283, #72240294, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 142640);