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 | |
6929 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,4,5)(3,6,8,7)(9,10,13,12)(11,15,16,14)(17,18,22,21)(19,23,20,24);(1,2)(3,7)(4,5)(6,8)(10,12)(14,15)(17,19,22,20)(18,23,21,24);(2,5)(6,7)(9,10,13,12)(11,14,16,15)(17,18,22,21)(19,24,20,23);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,15)(13,16)(17,19,22,20)(18,24,21,23);(1,2)(3,7)(4,5)(6,8)(9,11)(10,15)(12,14)(13,16)(17,20,22,19)(18,24,21,23);(1,3,4,8)(2,6,5,7)(9,12,13,10)(11,15,16,14)(17,21,22,18)(19,23,20,24)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=hi,d^2=gi,e^2=gi,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi101 conjugacy classes — view the full character table.
| maximal subgroup of | #72297124, #72297213, #72297217, … |
| maximal normal subgroup of | #72297124, #72297213, #72297217, … |
| maximal quotient of | #72297152, #72297230, #72297288, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7896922);