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 | |
6404 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,5,7)(3,4,6,8)(17,18,22,21)(19,20,24,23);(1,3,5,6)(2,4,7,8)(9,10)(11,14)(12,15)(13,16)(17,19)(18,23)(20,21)(22,24);(1,2)(3,8)(4,6)(5,7)(9,11)(10,13)(12,16)(14,15);(3,6)(4,8)(9,11)(10,14)(12,16)(13,15)(17,20,22,23)(18,19,21,24);(1,2)(3,4)(5,7)(6,8)(9,11)(10,14)(12,16)(13,15)(17,21,22,18)(19,20,24,23);(1,4)(2,6)(3,7)(5,8)(9,10)(11,13)(12,15)(14,16)(17,20)(18,24)(19,21)(22,23)Polycyclic presentation:
2^9|a^2=gh,b^2=h,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
| maximal subgroup of | #72232384, #72232519, #72234846, … |
| maximal normal subgroup of | #72232384, #72232519, #72234846, … |
| maximal quotient of | #72232407, #72232460, #72232662, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8564648);