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 | |
6652 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,6,5)(3,7,8,4)(17,18)(19,23)(20,22)(21,24);(1,3,6,8)(2,7,5,4)(9,10,15,16)(11,14,13,12)(17,19)(18,21)(20,24)(22,23);(2,5)(3,8)(9,11,15,13)(10,12,16,14)(18,22)(21,23);(1,2,6,5)(3,7,8,4)(9,12,15,14)(10,11,16,13)(17,18)(19,21)(20,22)(23,24);(1,4,6,7)(2,8,5,3)(9,13,15,11)(10,12,16,14)(17,19)(18,21)(20,24)(22,23);(1,5,6,2)(3,7,8,4)(9,14,15,12)(10,11,16,13)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=ghi,d^2=i,e^2=i,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,f^c=fi,e^d=ei98 conjugacy classes — view the full character table.
| maximal subgroup of | #72233839, #72233860, #72233992, … |
| maximal normal subgroup of | #72233839, #72233860, #72233992, … |
| maximal quotient of | #72233824, #72234032, #72234038, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7756859);