Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6343312 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×C22⋊C4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
3003 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,8,6,3,4,7)(10,15)(12,13)(16,22)(17,21)(19,20)(23,24);(1,3,6,2)(4,7,5,8)(9,10,14,17)(11,16,18,23)(12,15,13,21)(19,22,20,24);(9,11)(10,16)(12,19)(13,20)(14,18)(15,22)(17,23)(21,24);(11,18)(16,23)(19,20)(22,24);(9,12,14,13)(10,15,17,21)(11,19,18,20)(16,22,23,24)Polycyclic presentation:
2^9|a^2=g,b^2=i,e^2=h,f^2=i,g^2=hi,b^a=bf,e^a=eh,f^a=fh,f^b=fi,g^b=ghi,d^c=dh116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6251203);