Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153145 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13346 |
| commutator, Frattini, cosocle | C2×C4⋊Q8 | C23 |
| 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 | |
561 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 561 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:
(4,16)(5,21)(7,22)(9,26)(10,25)(11,29)(12,14)(13,31)(17,28)(18,32)(19,20)(27,30);(1,2)(3,11)(4,17)(5,10)(6,8)(7,13)(9,20)(12,30)(14,29)(15,27)(16,28)(18,22)(19,26)(21,25)(23,31)(24,32);(1,3)(2,7)(4,18)(5,12)(6,15)(8,22)(9,27)(10,23)(11,17)(13,20)(14,21)(16,32)(19,31)(24,25)(26,29)(28,30)Polycyclic presentation:
2^9|d^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fghi,g^a=gi,c^b=cf,d^b=dh,e^b=ei,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.
| maximal subgroup of | #72283075, #72283424 |
| maximal normal subgroup of | #72283075, #72283424 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60341);