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 | |
8863 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(9,10,13,12)(11,15,16,14)(17,18)(19,23)(20,21)(22,24);(1,2)(3,7)(4,5)(6,8)(9,11)(10,14)(12,15)(13,16)(17,18)(19,23)(20,21)(22,24);(2,5)(6,7)(11,16)(14,15)(18,21)(22,23);(17,18)(19,22)(20,21)(23,24);(1,3)(2,6)(4,8)(5,7)(9,12,13,10)(11,14,16,15)(17,19,20,24)(18,22,21,23);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,14)(13,16)(17,18)(19,22)(20,21)(23,24)Polycyclic presentation:
2^9|a^2=g,e^2=gh,b^a=bg,c^a=ch,e^a=eh,c^b=ci,e^b=ei,d^c=dh,f^c=fi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72232071, #72233566, #72233579, … |
| maximal normal subgroup of | #72232071, #72233566, #72233579, … |
| maximal quotient of | #72232158, #72232178, #72232298, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7652138);