Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #152606 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13049 |
| commutator | C4⋊Q8 | C22×C4 |
| Frattini, cosocle | #532 | C23 |
| prime radical | #3483 | C22 |
| LCS | G ▷ C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ G | |
| LECS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
453 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 453 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)(3,14,17,26)(4,19,22,27)(5,10,15,18)(6,13,8,11)(9,25)(16,24)(20,30)(21,29,32,31)(23,28);(1,3,15,2,6,18,26,13)(4,20,32,9,22,16,21,28)(5,11,8,17,14,12,7,10)(19,23,31,30,27,25,29,24);(1,4,6,22)(2,9,13,28)(3,16,18,20)(5,24,14,30)(7,23,8,25)(10,29,17,31)(11,27,12,19)(15,21,26,32)Polycyclic presentation:
2^9|a^2=f,b^2=df,c^2=i,d^2=i,e^2=h,g^2=i,b^a=bd,c^a=ce,e^a=egh,g^a=gh,h^a=hi,c^b=ch,d^b=di,e^b=egh,f^b=fi,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 | #72306593, #72306778, #72306779, … |
| maximal normal subgroup of | #72306593, #72306778, #72306779, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59803);