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 | |
5498 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,4,7)(3,5,8,6)(9,10)(11,12)(13,15)(14,16)(17,18)(19,23)(20,22)(21,24);(1,3)(2,5)(4,8)(6,7)(9,11,13,16)(10,14,15,12)(17,18,20,22)(19,21,24,23);(1,3,4,8)(2,6,7,5)(9,10)(11,14)(12,16)(13,15)(17,18)(19,21)(20,22)(23,24);(3,8)(5,6)(10,15)(11,16)(17,19)(18,21)(20,24)(22,23);(1,3)(2,5)(4,8)(6,7)(9,11)(10,12)(13,16)(14,15);(1,2)(3,5)(4,7)(6,8)(9,12)(10,11)(13,14)(15,16)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72255673, #72255696, #72256180, … |
| maximal normal subgroup of | #72255673, #72255696, #72256180, … |
| maximal quotient of | #72263324 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8965401);