Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #482362 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C22⋊C4 | C24 |
| prime radical | #35301 | C2 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1024 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)(3,11,13,16)(4,9)(5,10)(6,8)(7,12,15,14)(17,18)(19,23)(20,27,28,22)(21,24)(25,31)(26,30,32,29);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,11)(10,16)(18,22)(19,25)(23,30)(24,27)(26,32)(29,31);(2,8)(7,15)(9,10)(11,16)(17,19,21,25)(18,23,24,31)(20,26,28,32)(22,29,27,30);(1,4,6,5)(2,9,8,10)(3,12,13,14)(7,11,15,16)(17,20)(18,22)(19,26)(21,28)(23,29)(24,27)(25,32)(30,31)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=h,e^2=h,b^a=be,c^a=ch,d^a=dg,e^a=eg,g^a=gi,c^b=ci,e^b=eh,f^b=fgh,f^d=fi,f^e=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72284078, #72284113, #72284337, … |
| maximal normal subgroup of | #72284078, #72284113, #72284337, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 389753);