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 | |
6358 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)(3,8)(4,5)(6,7)(19,23)(21,24);(1,3,6,5)(2,4,7,8)(10,13)(14,16)(17,18)(19,24)(20,22)(21,23);(2,7)(4,8)(9,10,12,13)(11,14,15,16)(17,19,20,23)(18,21,22,24);(2,7)(3,5)(11,15)(14,16)(17,18,20,22)(19,21,23,24);(1,4)(2,3)(5,7)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,21)(20,23)(22,24);(1,5,6,3)(2,4,7,8)(9,11)(10,14)(12,15)(13,16)(17,18)(19,21)(20,22)(23,24)Polycyclic presentation:
2^9|b^2=g,c^2=i,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
| maximal subgroup of | #72249370, #72249465, #72249472, … |
| maximal normal subgroup of | #72249370, #72249465, #72249472, … |
| maximal quotient of | #72263090, #72263386 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8506213);