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 | C2×C4⋊Q8 | C23 |
| prime radical | #532 | D4 |
| LCS | G ▷ C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×C4⋊Q8 ◁ G | |
| LECS | G ▷ C2×C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
442 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 442 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:
(1,2,7,12,8,13,5,11)(3,14,17,9,18,24,16,6)(4,10,19,26,21,28,32,31)(15,22,29,20,27,25,30,23);(1,3)(2,9)(4,15,21,27)(5,17)(6,13)(7,16)(8,18)(10,23,28,20)(11,24)(12,14)(19,30,32,29)(22,26,25,31);(1,4)(2,10)(3,15)(5,22)(6,23)(7,25)(8,21)(9,20)(11,29)(12,30)(13,28)(14,19)(16,31)(17,26)(18,27)(24,32)Polycyclic presentation:
2^9|a^2=f,b^2=df,d^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,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.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59804);