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 | |
7663 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,4,6)(3,5,7,8)(9,10)(11,14)(12,15)(13,16)(17,18,21,23)(19,22,24,20);(3,7)(5,8)(9,11)(10,13)(12,16)(14,15)(17,19)(18,22)(20,23)(21,24);(1,2)(3,5)(4,6)(7,8)(17,18)(19,20)(21,23)(22,24);(9,10,12,15)(11,13,16,14)(17,20,21,22)(18,19,23,24);(1,3)(2,5)(4,7)(6,8)(9,10)(11,13)(12,15)(14,16)(17,20)(18,19)(21,22)(23,24);(1,2)(3,5)(4,6)(7,8)(9,11)(10,14)(12,16)(13,15)(17,18)(19,22)(20,24)(21,23)Polycyclic presentation:
2^9|a^2=h,d^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
| maximal subgroup of | #72232069, #72232276, #72232577, … |
| maximal normal subgroup of | #72232069, #72232276, #72232577, … |
| maximal quotient of | #72232093, #72232167, #72232389, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9130345);