Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5671 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2)(3,4)(5,7)(6,8)(11,16)(12,14)(17,18)(19,23)(20,21)(22,24);(1,3,6,7)(2,5,8,4)(9,10)(11,12)(13,15)(14,16)(18,21)(22,23);(9,11)(10,14)(12,15)(13,16)(17,18)(19,22)(20,21)(23,24);(1,4,6,5)(2,7,8,3)(9,10)(11,12)(13,15)(14,16)(17,19)(18,22)(20,24)(21,23);(1,5,6,4)(2,7,8,3)(9,10)(11,14)(12,16)(13,15);(1,2)(3,5)(4,7)(6,8)(9,12,13,14)(10,11,15,16)Polycyclic presentation:
2^9|b^2=ghi,d^2=ghi,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
| maximal subgroup of | #72256162, #72262887, #72262903, … |
| maximal normal subgroup of | #72256162, #72262887, #72262903, … |
| maximal quotient of | #72266262, #72266573, #72266578 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9772839);