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 | |
6097 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(3,4)(6,8)(17,18)(19,20)(21,23)(22,24);(1,2,5,7)(3,8,4,6)(9,10)(11,16)(12,14)(13,15)(17,19,22,21)(18,23,24,20);(1,3,5,4)(2,6,7,8)(9,11,13,12)(10,14,15,16);(1,2,5,7)(3,8,4,6)(9,10)(11,14)(12,16)(13,15)(17,20,22,23)(18,21,24,19);(1,2)(3,6)(4,8)(5,7)(9,12,13,11)(10,14,15,16)(17,21,22,19)(18,23,24,20);(1,4,5,3)(2,6,7,8)(9,10)(11,14)(12,16)(13,15)(17,18)(19,23)(20,21)(22,24)Polycyclic presentation:
2^9|b^2=gh,c^2=i,d^2=gh,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72262868, #72262945, #72263013, … |
| maximal normal subgroup of | #72262868, #72262945, #72263013, … |
| maximal quotient of | #72300749, #72300849 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10065313);