Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #152602 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13049 |
| commutator | #127 | C22×C4 |
| Frattini, cosocle | #532 | C23 |
| prime radical | #3481 | C22 |
| LCS | G ▷ #127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ G | |
| LECS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
458 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 458 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:
(1,2,5,11,7,3,8,14)(4,15,17,28,19,25,32,31)(6,13,23,29,22,10,21,27)(9,20,26,16,12,24,30,18);(1,3,5,14,7,2,8,11)(4,9,24,31,19,12,20,28)(6,13,21,27,22,10,23,29)(15,32,30,16,25,17,26,18);(1,4,7,19)(2,9,3,12)(5,20,8,24)(6,16,22,18)(10,25,13,15)(11,28,14,31)(17,21,32,23)(26,29,30,27)Polycyclic presentation:
2^9|a^2=f,b^2=df,c^2=i,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,e^a=egh,g^a=gh,h^a=hi,c^b=ch,e^b=egh,h^b=hi,d^c=dg,e^c=eh,f^c=fgi,g^c=gi,e^d=eh,g^d=gi,f^e=fh,g^f=gi38 conjugacy classes — view the full character table.
| maximal subgroup of | #72306625, #72306778, #72306779, … |
| maximal normal subgroup of | #72306625, #72306778, #72306779, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59801);