Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6632008 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2295 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,6)(3,4)(10,13)(11,15)(17,18)(19,21)(20,23)(22,24);(1,2,5,6)(3,8,4,7)(9,10,12,13)(11,14,15,16);(10,13)(14,16)(18,21)(19,24)(22,23);(1,3,5,4)(2,7,6,8)(9,11,12,15)(10,14,13,16);(1,4,5,3)(2,7,6,8)Polycyclic presentation:
2^9|b^2=f,d^2=f,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^b=dfh,g^c=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72282721, #72282945, #72283031 |
| maximal normal subgroup of | #72282721, #72282945, #72283031 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6562284);