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 | |
5296 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 24 points:
(3,4)(6,7)(9,10)(11,13)(12,16)(14,15)(17,18)(19,20)(21,22)(23,24);(1,2,5,8)(3,6,4,7)(9,11)(10,13)(12,14)(15,16)(17,19,23,21)(18,22,24,20);(1,3,5,4)(2,6,8,7)(9,10)(11,16)(12,13)(14,15)(18,24)(20,22);(10,15)(11,12)(17,20,23,22)(18,19,24,21);(1,2)(3,6)(4,7)(5,8)(9,12,14,11)(10,13,15,16)(17,21,23,19)(18,20,24,22);(1,4,5,3)(2,7,8,6)(9,13)(10,11)(12,15)(14,16)(17,22,23,20)(18,19,24,21)Polycyclic presentation:
2^9|b^2=h,c^2=gh,d^2=g,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8841965);