Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1714 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | #127 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#390 |
| prime radical | #60673 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1082 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,8,6,9,4,14)(3,10,13,15,11,16,12,7)(17,18)(19,26)(20,24)(21,30)(22,25)(23,28)(27,31)(29,32);(1,3)(2,7,9,15)(4,12)(5,13)(6,11)(8,10,14,16);(3,11)(7,15)(10,16)(12,13)(17,19,21,28,22,29,20,27)(18,23,24,26,25,31,30,32);(17,20,22,21)(18,24,25,30)(19,27,29,28)(23,26,31,32)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=f,d^2=i,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=ch104 conjugacy classes — view the full character table.
| maximal subgroup of | #72237263, #72237267, #72237274, … |
| maximal normal subgroup of | #72237263, #72237267, #72237274, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 113542);