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 | |
6699 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)(9,10)(11,12)(13,15)(14,16)(17,18)(19,21)(20,23)(22,24);(1,2,6,7)(3,5,4,8)(11,16)(12,14)(18,19)(22,23);(1,3,6,4)(2,5,7,8)(9,11,13,16)(10,14,15,12)(17,18,21,19)(20,22,24,23);(1,4,6,3)(2,5,7,8)(9,12)(10,11)(13,14)(15,16)(17,19,21,18)(20,23,24,22);(17,20,21,24)(18,22,19,23);(1,5)(2,4)(3,7)(6,8)(9,10)(11,14)(12,16)(13,15)Polycyclic presentation:
2^9|b^2=hi,c^2=h,d^2=g,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
| maximal subgroup of | #72233860 |
| maximal normal subgroup of | #72233860 |
| maximal quotient of | #72234024 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8350965);