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 | |
7242 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(3,7)(5,8)(10,14)(12,16)(17,18)(19,21)(20,23)(22,24);(1,2)(3,5)(4,6)(7,8)(9,10)(11,12)(13,14)(15,16)(18,19)(22,23);(1,2,4,6)(3,5,7,8)(9,11,13,15)(10,12,14,16)(17,18,21,19)(20,22,24,23);(1,2)(3,5)(4,6)(7,8)(9,12)(10,11)(13,16)(14,15)(17,19,21,18)(20,23,24,22);(1,2)(3,5)(4,6)(7,8)(9,11)(10,12)(13,15)(14,16)(17,20,21,24)(18,22,19,23);(1,3)(2,5)(4,7)(6,8)(9,10)(11,12)(13,14)(15,16)Polycyclic presentation:
2^9|c^2=ghi,d^2=h,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72232277, #72232817, #72256371, … |
| maximal normal subgroup of | #72232277, #72232817, #72256371, … |
| maximal quotient of | #72232377, #72232383, #72232583, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8034192);