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 | |
6225 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,5,7)(3,8,4,6)(9,10,13,14)(11,15,16,12);(3,4)(6,8)(9,11)(10,12)(13,16)(14,15)(17,18)(19,22)(20,23)(21,24);(1,3,5,4)(2,6,7,8)(9,10)(11,12)(13,14)(15,16)(17,19)(18,21)(20,24)(22,23);(3,4)(6,8)(10,14)(12,15)(17,18)(19,21)(20,23)(22,24);(1,4,5,3)(2,6,7,8)(9,12,13,15)(10,11,14,16)(17,19)(18,22)(20,24)(21,23);(1,2)(3,6)(4,8)(5,7)(9,11)(10,12)(13,16)(14,15)(17,18)(19,21)(20,23)(22,24)Polycyclic presentation:
2^9|a^2=h,c^2=gh,e^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 | #72336206 |
| maximal normal subgroup of | #72336206 |
| maximal quotient of | #72336463, #72336861 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9994916);