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 | |
6752 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(9,10)(11,16)(12,14)(13,15)(17,18)(19,21)(20,23)(22,24);(1,2)(3,8)(4,6)(5,7)(9,11,13,14)(10,12,15,16)(18,19)(22,23);(1,3,5,6)(2,4,7,8)(17,18,21,19)(20,22,24,23);(1,2)(3,4)(5,7)(6,8)(9,12,13,16)(10,11,15,14)(17,19,21,18)(20,23,24,22);(1,2)(3,4)(5,7)(6,8)(9,10)(11,12)(13,15)(14,16)(17,20)(18,22)(19,23)(21,24);(1,4)(2,6)(3,7)(5,8)(9,12)(10,14)(11,15)(13,16)Polycyclic presentation:
2^9|b^2=gh,c^2=i,d^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 | #72232195, #72232576, #72234847, … |
| maximal normal subgroup of | #72232195, #72232576, #72234847, … |
| maximal quotient of | #72232249, #72232494, #72232637, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8571711);