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 | |
7627 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)(4,6)(7,8)(9,10,13,15)(11,14,16,12);(1,2,5,3)(4,7,8,6)(9,11)(10,14)(12,15)(13,16)(17,18)(19,23)(20,21)(22,24);(2,3)(6,7)(10,15)(11,16)(18,21)(22,23);(17,18)(19,22)(20,21)(23,24);(1,3,5,2)(4,7,8,6)(9,11)(10,12)(13,16)(14,15)(17,18)(19,22)(20,21)(23,24);(1,4)(2,6)(3,7)(5,8)(9,12)(10,11)(13,14)(15,16)(17,19)(18,22)(20,24)(21,23)Polycyclic presentation:
2^9|a^2=gh,b^2=g,e^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72232069, #72232197, #72232285, … |
| maximal normal subgroup of | #72232069, #72232197, #72232285, … |
| maximal quotient of | #72232131, #72232159, #72232220, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7640357);