Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139424 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #9428 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C4×M16 | C23 |
| prime radical | #19750 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ G | |
| LECS | G ▷ C4×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
353 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 353 conjugacy classes of subgroups — too many to draw. Only its 60 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 — 3 generators on 32 points:
(1,2,7,12,21,11,5,10,6,13,22,23,8,3,14,16)(4,15,19,30,25,28,24,32,18,29,31,26,20,9,17,27);(1,3,8,2,6,11,21,13)(4,15,25,9,18,29,20,28)(5,10,22,12,14,16,7,23)(17,30,31,32,24,26,19,27);(1,4)(2,9)(3,15)(5,19)(6,18)(7,24)(8,25)(10,27)(11,29)(12,30)(13,28)(14,31)(16,32)(17,22)(20,21)(23,26)Polycyclic presentation:
2^9|a^2=f,b^2=dfhi,d^2=gh,e^2=gi,f^2=gh,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,d^b=dh,e^b=egh,f^b=fhi,g^b=gi,d^c=dgh,e^c=egi,f^c=fgh,h^c=hi,f^d=fi38 conjugacy classes — view the full character table.
| maximal subgroup of | #72306588, #72306765, #72306767, … |
| maximal normal subgroup of | #72306588, #72306765, #72306767, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 46622);