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 | |
5518 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,5,7)(3,6,8,4)(17,18,22,23)(19,21,20,24);(1,3)(2,6)(4,7)(5,8)(9,10)(11,16)(12,14)(13,15)(18,23)(19,20);(1,2)(3,4)(5,7)(6,8)(9,11,13,14)(10,12,15,16)(17,19,22,20)(18,24,23,21);(3,8)(4,6)(9,10)(11,12)(13,15)(14,16)(17,20,22,19)(18,21,23,24);(1,2)(3,6)(4,8)(5,7)(9,10)(11,12)(13,15)(14,16)(17,21)(18,19)(20,23)(22,24);(1,4,5,6)(2,3,7,8)(9,12)(10,14)(11,15)(13,16)Polycyclic presentation:
2^9|a^2=h,c^2=ghi,d^2=g,f^2=gh,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 | #72256306, #72262945, #72262957 |
| maximal normal subgroup of | #72256306, #72262945, #72262957 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8572551);