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 | |
6316 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(3,8)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18)(19,22)(20,23)(21,24);(1,2)(3,7)(4,6)(5,8)(17,19)(18,21)(20,24)(22,23);(1,3,4,8)(2,5,6,7)(9,10,12,14)(11,13,15,16)(17,18)(19,22)(20,23)(21,24);(2,6)(5,7)(11,15)(13,16)(17,19,20,24)(18,22,23,21);(1,3)(2,7)(4,8)(5,6)(9,11)(10,13)(12,15)(14,16)(17,19)(18,21)(20,24)(22,23);(1,2)(3,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,21)(20,23)(22,24)Polycyclic presentation:
2^9|c^2=h,d^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 | #72232280, #72232285, #72234803, … |
| maximal normal subgroup of | #72232280, #72232285, #72234803, … |
| maximal quotient of | #72232313, #72232335, #72232422, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7984541);