Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7943 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#95 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C2×#336 |
| prime radical | C24 | C22⋊C8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
451 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,9)(4,12)(5,8)(6,11)(7,13)(10,15)(14,16)(17,18,20,25,21,23,29,30,22,26,28,32,19,24,27,31);(1,3)(2,6,8,14)(4,11)(5,10)(7,9,12,15)(13,16)(18,23,26,24)(20,28)(25,31,32,30)(27,29);(3,10)(6,14)(9,15)(11,16)Polycyclic presentation:
2^9|a^2=e,b^2=f,d^2=gi,e^2=h,h^2=i,b^a=bd,d^a=dg,f^a=fgi,c^b=cg,e^b=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72285467, #72285490, #72285533, … |
| maximal normal subgroup of | #72285467, #72285490, #72285533, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 33831);