Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #137497 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1879 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #33748 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1451 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
343 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 343 conjugacy classes of subgroups — too many to draw. Only its 76 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) = 256 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,6,8,11)(3,9,10,12)(17,18,21,25,22,27,19,23)(20,29,31,32,30,24,28,26);(1,2,4,11,5,8,13,6)(3,7,12,15,10,16,9,14)(18,23,27,25)(20,30)(24,26,29,32)(28,31);(1,3)(2,7)(4,12)(5,10)(6,14)(8,16)(9,13)(11,15)(18,24)(19,21)(20,31)(23,26)(25,32)(27,29)(28,30)Polycyclic presentation:
2^9|a^2=f,b^2=df,d^2=gi,e^2=gh,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,h^a=hi,d^b=dg,e^b=egi,f^b=fg,d^c=dgi,e^c=eghi,f^c=fgi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72225273, #72227102, #72227159, … |
| maximal normal subgroup of | #72225273, #72227102, #72227159, … |
| maximal quotient of | #72225244 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44697);