Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #138698 |
| 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 | #33675 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
349 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 349 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) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,8,6,9,4,14)(3,10,13,15,11,16,12,7)(17,18,22,26,32,28,21,29,23,20,30,19,24,27,31,25);(1,3)(2,7,9,15)(4,12)(5,13)(6,11)(8,10,14,16)(17,19,24,25,23,26,32,29)(18,21,28,22,20,31,27,30);(3,11)(7,15)(10,16)(12,13)(17,20)(18,23)(19,21)(22,25)(24,27)(26,31)(28,32)(29,30)Polycyclic presentation:
2^9|a^2=f,b^2=dfhi,d^2=ghi,e^2=hi,f^2=gh,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,c^b=cg,d^b=dh,e^b=ehi,f^b=fh,h^b=hi,d^c=dhi,e^c=eh,f^c=fhi,h^c=hi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72287688, #72287703, #72287708, … |
| maximal normal subgroup of | #72287688, #72287703, #72287708, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 45959);