Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #141751 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1957 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #19603 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
367 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 367 conjugacy classes of subgroups — too many to draw. Only its 102 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:
(1,2,4,7,6,10,5,9)(3,11,13,16,12,15,14,8)(18,22,24,27)(19,26)(20,29)(23,28)(25,32,31,30);(2,7,10,9)(3,12)(8,15,16,11)(13,14)(17,18,23,30,29,31,28,22,21,24,26,32,20,25,19,27);(1,3,5,14,6,12,4,13)(2,8,7,11,10,16,9,15)Polycyclic presentation:
2^9|a^2=f,b^2=dfgi,c^2=e,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,c^b=ch,d^b=di,f^b=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72213965, #72213975, #72214020, … |
| maximal normal subgroup of | #72213965, #72213975, #72214020, … |
| maximal quotient of | #72216171, C3×#141837, #72226070, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 49033);