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 | C2×#121 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4776 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 24 points:
(3,7)(4,8)(10,13)(14,16)(17,18,21,20)(19,23,24,22);(1,2,5,6)(3,8,7,4)(9,10)(11,14)(12,13)(15,16);(1,3,5,7)(2,4,6,8)(17,19,21,24)(18,22,20,23);(1,2,5,6)(3,8,7,4)(17,20,21,18)(19,22,24,23);(1,2)(3,4)(5,6)(7,8)(9,11,12,15)(10,14,13,16);(1,4)(2,3)(5,8)(6,7)Polycyclic presentation:
2^9|a^2=hi,b^2=i,c^2=h,d^2=h,e^2=g,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=dh,f^c=fi,e^d=ei110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7589739);