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 |
| prime radical | #36644 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5854 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 24 points:
(2,6)(4,7)(9,10,15,13)(11,12,16,14)(19,23)(22,24);(2,6)(3,8)(10,13)(11,16)(17,18,21,20)(19,22,23,24);(1,2,5,6)(3,7,8,4)(9,11)(10,14)(12,13)(15,16)(17,19,21,23)(18,22,20,24);(2,6)(3,8)(9,12,15,14)(10,16,13,11);(1,3)(2,7)(4,6)(5,8)(9,13,15,10)(11,12,16,14);(1,4,5,7)(2,3,6,8)(9,14,15,12)(10,16,13,11)(17,20,21,18)(19,24,23,22)Polycyclic presentation:
2^9|a^2=g,b^2=hi,c^2=gh,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=di,e^d=ei128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7538347);