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 | |
5589 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(9,10,14,12)(11,15,13,16)(17,18)(19,20)(21,22)(23,24);(1,2)(3,6)(4,7)(5,8)(9,11,14,13)(10,15,12,16)(18,22)(19,23);(1,3)(2,5)(4,8)(6,7)(9,11,14,13)(10,16,12,15)(17,19,21,23)(18,20,22,24);(3,8)(5,6)(9,12,14,10)(11,15,13,16)(17,20)(18,23)(19,22)(21,24);(1,2)(3,5)(4,7)(6,8)(9,13,14,11)(10,15,12,16);(1,3)(2,6)(4,8)(5,7)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=gh,d^2=h,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
| maximal subgroup of | #72256306, #72262938, #72262949, … |
| maximal normal subgroup of | #72256306, #72262938, #72262949, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8610798);