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 | |
6386 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)(3,5)(4,7)(6,8)(9,10)(11,13)(12,15)(14,16)(18,21)(19,20);(1,2,5,3)(4,6,8,7)(9,11)(10,12)(13,16)(14,15)(17,18,22,21)(19,24,20,23);(1,2,5,3)(4,6,8,7)(9,12,14,13)(10,15,16,11)(17,19,22,20)(18,23,21,24);(2,3)(6,7)(10,16)(12,13)(17,20,22,19)(18,24,21,23);(1,3,5,2)(4,7,8,6)(9,13,14,12)(10,11,16,15)(17,21,22,18)(19,24,20,23);(1,4)(2,6)(3,7)(5,8)(9,11)(10,12)(13,16)(14,15)Polycyclic presentation:
2^9|b^2=h,c^2=g,d^2=i,e^2=g,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^d=egi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72326662 |
| maximal normal subgroup of | #72326662 |
| maximal quotient of | #72326708, #72327462, #72328130, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8066549);