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 | |
7172 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2)(3,8)(4,6)(5,7)(9,10,15,16)(11,12,13,14);(2,4)(7,8)(9,11)(10,12)(13,15)(14,16)(17,18,23,24)(19,21,22,20);(1,3,6,5)(2,7,4,8)(9,12,15,14)(10,11,16,13)(17,19,23,22)(18,20,24,21);(1,2,6,4)(3,7,5,8)(9,11,15,13)(10,12,16,14)(17,20,23,21)(18,19,24,22);(1,4,6,2)(3,7,5,8)(9,13,15,11)(10,12,16,14)(17,21,23,20)(18,19,24,22);(1,5,6,3)(2,7,4,8)(9,14,15,12)(10,11,16,13)(17,22,23,19)(18,20,24,21)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=gi,d^2=gi,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
| maximal subgroup of | #72233550, #72233561, #72233609, … |
| maximal normal subgroup of | #72233550, #72233561, #72233609, … |
| maximal quotient of | #72234011, #72234013, #72234036, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8564335);