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 | |
6362 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,5,4)(3,6,8,7)(17,18)(19,21)(20,23)(22,24);(2,4)(6,7)(9,10,13,14)(11,15,16,12)(18,19)(22,23);(1,3,5,8)(2,6,4,7)(10,14)(12,15)(17,18,21,19)(20,22,24,23);(2,4)(6,7)(9,11,13,16)(10,15,14,12)(17,19,21,18)(20,23,24,22);(1,4,5,2)(3,7,8,6)(9,11)(10,12)(13,16)(14,15)(17,20,21,24)(18,22,19,23);(1,3)(2,6)(4,7)(5,8)(9,12)(10,11)(13,15)(14,16)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,c^2=g,d^2=i,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72232200, #72238561, #72238565, … |
| maximal normal subgroup of | #72232200, #72238561, #72238565, … |
| maximal quotient of | #72232271, #72232537, #72232539, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8265033);