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 | |
6264 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2)(3,5)(4,8)(6,7)(9,10,12,13)(11,15,16,14)(17,18,22,20)(19,23,21,24);(2,7)(3,4)(10,13)(14,15)(17,19,22,21)(18,23,20,24);(2,7)(5,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,21)(18,24,20,23);(1,3,6,4)(2,8,7,5)(9,10)(11,14)(12,13)(15,16)(17,20,22,18)(19,23,21,24);(1,4,6,3)(2,5,7,8)(9,11,12,16)(10,14,13,15)(17,21,22,19)(18,23,20,24);(1,5)(2,3)(4,7)(6,8)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=gh,d^2=hi,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72232446, #72249674, #72262922, … |
| maximal normal subgroup of | #72232446, #72249674, #72262922, … |
| maximal quotient of | #72232494, #72263198, #72300351, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8118904);