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 | |
6089 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(2,3)(6,7)(11,16)(13,15)(17,18)(19,21)(20,22)(23,24);(1,2,5,3)(4,6,8,7)(9,10)(11,15)(12,14)(13,16);(2,3)(6,7)(9,11,12,16)(10,13,14,15)(17,18,20,22)(19,21,23,24);(2,3)(4,8)(10,14)(13,15)(19,23)(21,24);(1,3,5,2)(4,7,8,6)(9,11)(10,15)(12,16)(13,14)(17,19)(18,21)(20,23)(22,24);(1,4)(2,6)(3,7)(5,8)(9,10)(11,15)(12,14)(13,16)(17,18)(19,21)(20,22)(23,24)Polycyclic presentation:
2^9|b^2=g,c^2=h,e^2=g,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di,e^c=ei,f^c=fh,e^d=ehi,f^d=fi95 conjugacy classes — view the full character table.
| maximal subgroup of | #72256120, #72262940, #72262959, … |
| maximal normal subgroup of | #72256120, #72262940, #72262959, … |
| maximal quotient of | #72263193, #72263459 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7984550);