Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11753 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C22⋊C4 | C24 |
| socle | C22 | C22×#99 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
964 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,11,13,16)(4,9)(5,10)(6,8)(7,12,15,14)(17,18)(19,27)(20,29,31,24)(21,25)(22,26)(23,28,32,30);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,11)(10,16)(17,19,22,21)(18,23,26,32)(20,28,31,30)(24,27,29,25);(2,8)(7,15)(9,10)(11,16)(17,20)(18,24)(19,28)(21,30)(22,31)(23,27)(25,32)(26,29);(1,4,6,5)(2,9,8,10)(3,12,13,14)(7,11,15,16)(17,21,22,19)(18,25,26,27)(20,30,31,28)(23,24,32,29)Polycyclic presentation:
2^9|a^2=f,b^2=i,d^2=hi,e^2=h,b^a=be,c^a=cgh,e^a=eg,g^a=gi,e^b=eh,f^b=fgh,f^c=fi,f^e=fi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72284994, #72285155, #72285186, … |
| maximal normal subgroup of | #72284994, #72285155, #72285186, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 389432);