Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #94417 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7302 |
| commutator | C2×C16 | C42 |
| Frattini, cosocle | C4×M32 | C22 |
| prime radical | C23 | #360 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×C4×C8 ◁ C4×M32 ◁ G | |
| LECS | G ▷ C4×M32 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
134 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 134 conjugacy classes of subgroups — too many to draw. Only its 50 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(2,9,10,16,14,27,13,28)(3,15,18,4)(5,24)(6,7)(11,29,30,25,32,31,26,12)(17,21)(19,22,20,23);(1,2,4,11,21,31,18,28,6,10,22,26,17,12,20,27,8,14,15,32,24,29,3,16,7,13,23,30,5,25,19,9)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fi,d^2=gh,e^2=fg,f^2=h,g^2=i,h^2=i,b^a=bc,c^a=cf,e^a=efgi,f^a=fh,g^a=gi,h^a=hi,c^b=cg,d^b=dh,f^b=fi,g^b=gi,d^c=di,e^d=ei50 conjugacy classes — view the full character table.
| maximal subgroup of | #72214546, #72214585, #72214589, … |
| maximal normal subgroup of | #72214546, #72214585, #72214589, … |
| maximal quotient of | #72216302 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1613);