Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×#447 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3223 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,8)(4,6)(5,7)(9,10,12,14)(11,13,15,16)(17,18,20,23)(19,21,24,22);(1,3,5,4)(2,6,7,8)(9,11)(10,13)(12,15)(14,16);(11,15)(13,16)(18,21)(19,24)(22,23);(1,4,5,3)(2,6,7,8)(9,11)(10,13)(12,15)(14,16);(11,15)(13,16)Polycyclic presentation:
2^9|a^2=hi,b^2=f,d^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,d^c=dh,g^c=gi,e^d=eh95 conjugacy classes — view the full character table.
| maximal subgroup of | #72256896, #72256901, #72256987, … |
| maximal normal subgroup of | #72256896, #72256901, #72256987, … |
| maximal quotient of | #72257855, #72264260, #72305835, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6500064);