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 | |
6165 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,5,4)(3,7,8,6)(11,16)(13,15);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,14)(13,16)(17,18)(19,22)(20,23)(21,24);(9,11)(10,13)(12,16)(14,15)(17,19)(18,21)(20,24)(22,23);(2,4)(6,7)(10,14)(13,15)(17,19)(18,22)(20,24)(21,23);(1,3)(2,7)(4,6)(5,8)(9,11)(10,15)(12,16)(13,14)(17,19)(18,22)(20,24)(21,23);(1,4,5,2)(3,7,8,6)(9,10)(11,13)(12,14)(15,16)(17,18)(19,21)(20,23)(22,24)Polycyclic presentation:
2^9|a^2=g,f^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 | #72327643, #72327647, #72336777 |
| maximal normal subgroup of | #72327643, #72327647, #72336777 |
| maximal quotient of | #72327689, #72327735, #72327865, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8564469);