Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
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 | |
7071 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)(9,10,14,13)(11,16,12,15)(17,18)(19,20)(21,23)(22,24);(1,2)(3,7)(4,5)(6,8)(10,13)(15,16)(17,19)(18,20)(21,24)(22,23);(2,5)(6,7)(9,10,14,13)(11,15,12,16)(17,20,22,21)(18,19,24,23);(1,2)(3,6)(4,5)(7,8)(9,11,14,12)(10,15,13,16)(17,19)(18,21)(20,24)(22,23);(1,2)(3,7)(4,5)(6,8)(9,12,14,11)(10,15,13,16)(17,19)(18,20)(21,24)(22,23);(1,3,4,8)(2,6,5,7)(9,13,14,10)(11,16,12,15)(17,21,22,20)(18,23,24,19)Polycyclic presentation:
2^9|a^2=g,c^2=hi,d^2=gh,e^2=gh,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 | #72236189, #72236627, #72262811, … |
| maximal normal subgroup of | #72236189, #72236627, #72262811, … |
| maximal quotient of | #72262821, #72297397, #72297416, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7892490);