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 | |
6400 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(2,5)(3,7)(10,13)(14,16)(18,21)(22,23);(2,5)(6,8)(17,18,20,21)(19,22,24,23);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16);(3,7)(6,8)(11,15)(14,16)(17,18,20,21)(19,23,24,22);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,15)(13,16)(17,18)(19,22)(20,21)(23,24);(1,3)(2,6)(4,7)(5,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,22)(20,24)(21,23)Polycyclic presentation:
2^9|b^2=g,d^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
| maximal subgroup of | #72232280, #72232833, #72263506, … |
| maximal normal subgroup of | #72232280, #72232833, #72263506, … |
| maximal quotient of | #72232383, #72232399, #72232892, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10315071);