Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11541 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#99 |
| commutator | C24 | C2×C42 |
| Frattini, cosocle | C23×D4 | C23 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C25 ◁ C24×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C22 ▷ C2 ▷ C1 | |
2664 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2)(3,4)(5,6)(7,8)(9,10,13,16,14,11,12,15);(2,4)(9,11,14,10)(12,16)(13,15);(6,8)Polycyclic presentation:
2^9|a^2=f,b^2=g,f^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gh,h^a=hi,d^b=dh,f^b=fh,h^b=hi,f^d=fi,g^d=gi,g^f=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72250152, #72250195, #72280502, … |
| maximal normal subgroup of | #72250152, #72250195, #72280502, … |
| maximal quotient of | #72250038, #72250648, #72250820, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53762);