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 | |
7193 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(9,10)(11,12)(13,16)(14,15)(17,18,20,21)(19,23,24,22);(2,5)(6,8)(9,11)(10,12)(13,14)(15,16)(18,21)(22,23);(1,2,4,5)(3,6,7,8)(9,12)(10,13)(11,15)(14,16)(17,18)(19,22)(20,21)(23,24);(3,7)(6,8)(9,11,14,13)(10,12,15,16)(17,18)(19,22)(20,21)(23,24);(1,2)(3,6)(4,5)(7,8)(9,13,14,11)(10,12,15,16)(17,19,20,24)(18,22,21,23);(1,3)(2,6)(4,7)(5,8)(9,12)(10,11)(13,15)(14,16)Polycyclic presentation:
2^9|a^2=g,c^2=hi,d^2=gh,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
| maximal subgroup of | #72232068, #72232069, #72233993, … |
| maximal normal subgroup of | #72232068, #72232069, #72233993, … |
| maximal quotient of | #72232146, #72232147, #72232514, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8507286);