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 | #59403 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4958 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,5,4)(3,7,8,6)(9,10,13,16)(11,14,12,15)(17,18)(19,20)(21,23)(22,24);(1,2,5,4)(3,7,8,6)(9,10)(11,14)(12,15)(13,16)(17,19,22,21)(18,20,24,23);(1,3)(2,6)(4,7)(5,8)(9,11,13,12)(10,14,16,15)(17,19,22,21)(18,23,24,20);(1,2,5,4)(3,7,8,6)(17,20,22,23)(18,19,24,21);(1,4,5,2)(3,7,8,6)(9,11,13,12)(10,14,16,15)(17,21,22,19)(18,23,24,20);(1,3)(2,7)(4,6)(5,8)(9,12,13,11)(10,15,16,14)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=ghi,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,c^b=ci,d^c=dh,f^e=fi113 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7598694);