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 |
| prime radical | C2×D42 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5391 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,5,6)(3,7,8,4)(9,10,13,14)(11,12,15,16)(17,18,20,22)(19,23,24,21);(2,6)(3,8)(17,19,20,24)(18,21,22,23);(1,3)(2,7)(4,6)(5,8)(10,14)(11,15);(1,4)(2,8)(3,6)(5,7)(9,11)(10,12)(13,15)(14,16);(1,4)(2,3)(5,7)(6,8)(9,12)(10,11)(13,16)(14,15);(1,3)(2,7)(4,6)(5,8)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,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.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7876570);