Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C211×C4 |
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 | |
7003 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(2,7)(3,4)(10,11)(14,15);(1,2)(3,5)(4,8)(6,7)(9,10,13,11)(12,14,16,15)(17,18,22,21)(19,24,20,23);(1,3,6,4)(2,5,7,8)(17,19,22,20)(18,23,21,24);(1,3,6,4)(2,8,7,5)(9,11,13,10)(12,14,16,15)(17,20,22,19)(18,23,21,24);(1,4,6,3)(2,8,7,5);(1,5,6,8)(2,3,7,4)(9,12,13,16)(10,14,11,15)(17,21,22,18)(19,24,20,23)Polycyclic presentation:
2^9|b^2=ghi,c^2=hi,d^2=gi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
| maximal subgroup of | #72238563, #72238580 |
| maximal normal subgroup of | #72238563, #72238580 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8336770);