Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #123071 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1595 |
| commutator, Frattini, cosocle | C2×C4⋊C8 | C23 |
| prime radical | #10254 | C2 |
| LCS | G ▷ C2×C4⋊C8 ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C8 ◁ G | |
| LECS | G ▷ C2×C4⋊C8 ▷ C22×C4 ▷ C22 ▷ C1 | |
343 conjugacy classes of subgroups; only the 44 normal subgroups are shown.
This group has 343 conjugacy classes of subgroups — too many to draw. Only its 44 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) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,7)(3,6)(4,11)(8,16)(9,15)(10,13)(12,14)(17,18)(19,22)(20,23)(21,25)(24,29)(26,31)(27,32)(28,30);(1,2,5,9)(3,8,6,14)(4,12,13,16)(7,10,15,11)(17,19)(18,22)(20,28)(21,26)(23,32)(24,30)(25,31)(27,29);(1,3,5,6)(2,8,9,14)(4,10,13,11)(7,12,15,16)(18,23)(19,26)(20,29)(22,30)(24,25)(31,32)Polycyclic presentation:
2^9|b^2=h,c^2=h,d^2=fh,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=df,e^a=ei,f^a=fh,g^a=gi,c^b=ci,d^b=df,e^b=egi,f^b=fh,d^c=dg,e^c=ei,e^d=ei41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30270);