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 | |
6896 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)(3,8)(4,7)(5,6)(9,10,14,15)(11,12,13,16);(1,2)(3,4)(5,6)(7,8)(10,15)(11,13)(17,18)(19,24)(20,21)(22,23);(2,6)(3,7)(9,11,14,13)(10,16,15,12)(18,23)(19,20);(17,19,22,20)(18,24,23,21);(1,3,5,7)(2,4,6,8)(9,12,14,16)(10,11,15,13)(17,20,22,19)(18,21,23,24);(1,4)(2,7)(3,6)(5,8)(9,13,14,11)(10,16,15,12)(17,21)(18,19)(20,23)(22,24)Polycyclic presentation:
2^9|a^2=g,c^2=g,d^2=hi,e^2=i,f^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 | #72297638, #72342309, #72342312 |
| maximal normal subgroup of | #72297638, #72342309, #72342312 |
| maximal quotient of | #72297653, #72297673, #72297928 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7644616);