Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153191 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13348 |
| commutator, Frattini, cosocle | C2×C4⋊Q8 | C23 |
| prime radical | C2×#2204 | C2 |
| LCS | G ▷ C2×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 | |
510 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 510 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 — 3 generators on 32 points:
(1,2)(3,11,15,7)(4,16)(5,20)(6,8)(9,19)(10,22)(12,29,18,32)(13,24,14,25)(17,28)(21,26)(23,30,27,31);(3,12)(4,17)(7,23)(9,26)(11,27)(13,31)(14,30)(15,18)(16,28)(19,21)(24,29)(25,32);(1,3)(2,7)(4,18)(5,13)(6,15)(8,11)(9,27)(10,24)(12,17)(14,22)(16,29)(19,31)(20,25)(21,30)(23,26)(28,32)Polycyclic presentation:
2^9|a^2=h,d^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fgh,g^a=gi,c^b=cf,d^b=dh,f^b=fi,g^b=gh,d^c=dg,f^c=fi,g^c=gi,h^c=hi,f^d=fh,g^f=gi35 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60387);