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 | |
6190 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,6,4)(3,5,8,7)(10,14)(12,16);(1,3)(2,7)(4,5)(6,8)(10,14)(11,15)(17,18)(19,20)(21,23)(22,24);(9,10)(11,12)(13,14)(15,16)(17,19)(18,22)(20,23)(21,24);(2,4)(5,7)(11,15)(12,16)(17,20,21,22)(18,19,23,24);(1,4,6,2)(3,5,8,7)(9,11,13,15)(10,12,14,16)(17,19)(18,20)(21,24)(22,23);(1,5)(2,8)(3,4)(6,7)(9,12)(10,11)(13,16)(14,15)(17,18)(19,22)(20,24)(21,23)Polycyclic presentation:
2^9|a^2=g,d^2=hi,e^2=gh,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 | #72256120, #72262986, #72262997, … |
| maximal normal subgroup of | #72256120, #72262986, #72262997, … |
| maximal quotient of | #72263218, #72327416, #72327758, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9131028);