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 | |
5575 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)(3,7)(4,6)(5,8)(9,10,12,14)(11,13,15,16)(18,21)(19,23);(9,10)(11,13)(12,14)(15,16)(18,21)(22,24);(1,3)(2,5)(4,8)(6,7)(17,18,20,21)(19,22,23,24);(1,2,4,6)(3,5,8,7)(17,19,20,23)(18,22,21,24);(1,2)(3,5)(4,6)(7,8);(1,3)(2,5)(4,8)(6,7)(9,11,12,15)(10,13,14,16)Polycyclic presentation:
2^9|a^2=g,c^2=i,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^d=ehi98 conjugacy classes — view the full character table.
| maximal subgroup of | #72262962, #72263011, #72263029 |
| maximal normal subgroup of | #72262962, #72263011, #72263029 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7552543);