Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #129554 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#102 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#466 | C4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
508 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 508 conjugacy classes of subgroups — too many to draw. Only its 114 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2)(3,4)(6,10,11,12)(13,14,16,21,17,22,26,19,18,23,15,20,25,28,24,27);(5,6,7,10,9,11,8,12)(14,19,23,27)(15,24)(16,26)(17,25)(20,22,21,28);(2,4)(5,7,9,8)(6,10,11,12)(13,15,17,24,18,16,25,26)(14,20,22,27,23,21,28,19)Polycyclic presentation:
2^9|a^2=f,b^2=dfgi,c^2=g,d^2=g,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,c^b=ch,d^b=dg,f^b=fg,g^b=gi,h^b=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72225030, #72225053, #72225063, … |
| maximal normal subgroup of | #72225030, #72225053, #72225063, … |
| maximal quotient of | #72225021, #72225024, #72225027, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43837);