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 | |
7759 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,5,6)(3,7,8,4)(9,10,13,16)(11,12,15,14)(17,18,20,22)(19,23,24,21);(1,3)(2,4)(5,8)(6,7)(9,11)(10,14)(12,16)(13,15)(17,19,20,24)(18,21,22,23);(9,10)(11,14)(12,15)(13,16)(17,19)(18,21)(20,24)(22,23);(2,6)(4,7)(9,10)(11,12)(13,16)(14,15)(17,19)(18,21)(20,24)(22,23);(1,3)(2,4)(5,8)(6,7)(17,18)(19,21)(20,22)(23,24);(1,4)(2,3)(5,7)(6,8)(9,12)(10,15)(11,16)(13,14)Polycyclic presentation:
2^9|a^2=g,b^2=i,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 | #72232195, #72232200, #72232275, … |
| maximal normal subgroup of | #72232195, #72232200, #72232275, … |
| maximal quotient of | #72232223, #72232298, #72232464, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8340870);