Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #136366 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1853 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#41 | SD16 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C8×#41 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
272 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 272 conjugacy classes of subgroups — too many to draw. Only its 84 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:
(1,2,6,9,7,4,5,10)(3,11,13,8,15,16,14,12)(18,22,23,30)(19,25)(20,28)(24,31,29,26)(27,32);(1,3,5,14,7,15,6,13)(2,8,10,11,4,12,9,16)(17,18,20,29,21,23,28,24)(19,26,25,22,27,31,32,30);(1,4)(2,7)(3,12)(5,9)(6,10)(8,15)(11,13)(14,16)Polycyclic presentation:
2^9|a^2=f,b^2=fg,d^2=g,e^2=h,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,d^b=dg,e^b=eh,f^b=fi,g^b=gi,d^c=dh,e^c=eh62 conjugacy classes — view the full character table.
| maximal subgroup of | #72225274, #72227118, #72227161, … |
| maximal normal subgroup of | #72225274, #72227118, #72227161, … |
| maximal quotient of | #72225151 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43658);