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 | C2×2−1+4 | D4 |
| 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 | |
494 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 494 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)(3,8)(4,14,18,9)(5,19)(6,23)(7,10)(11,13)(12,28,16,24)(15,30,31,27)(17,26,21,32)(20,29)(22,25);(1,3,7,6)(2,8,10,23)(4,15,18,31)(5,20,22,13)(9,26,14,32)(11,19,29,25)(12,21,16,17)(24,30,28,27);(1,4)(2,9)(3,12)(5,21)(6,16)(7,18)(8,24)(10,14)(11,30)(13,31)(15,20)(17,22)(19,26)(23,28)(25,32)(27,29)Polycyclic presentation:
2^9|a^2=h,b^2=i,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, 60388);