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 | |
6393 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,6,8)(3,4,5,7)(19,23)(20,24);(1,3,6,5)(2,7,8,4)(9,10)(11,12)(13,15)(14,16)(17,18,22,21)(19,24,23,20);(1,3,6,5)(2,7,8,4)(9,11)(10,14)(12,15)(13,16)(17,19)(18,20)(21,24)(22,23);(2,8)(3,5)(11,16)(12,14)(18,21)(19,23);(1,4)(2,3)(5,8)(6,7)(9,10)(11,14)(12,16)(13,15)(17,20)(18,19)(21,23)(22,24);(1,5,6,3)(2,7,8,4)(9,12,13,14)(10,11,15,16)(17,21,22,18)(19,24,23,20)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
| maximal subgroup of | #72263262 |
| maximal normal subgroup of | #72263262 |
| maximal quotient of | #72263309 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8628391);