Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1665 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #121 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | #380 |
| prime radical | C22×D4 | Q8 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
233 conjugacy classes of subgroups; only the 77 normal subgroups are shown.
This group has 233 conjugacy classes of subgroups — too many to draw. Only its 77 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,5)(4,6)(7,8)(9,10,13,16,14,17,12,22)(11,18,20,23,21,24,19,15);(2,5)(6,7)(9,11,12,19,14,21,13,20)(10,15,16,18,17,23,22,24);(2,6)(4,8)(5,7)Polycyclic presentation:
2^8|a^2=f,b^2=dh,d^2=g,e^2=h,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,e^c=eh58 conjugacy classes — view the full character table.
| maximal subgroup of | #131712, #131713, #131720, … |
| maximal normal subgroup of | #131712, #131713, #131720, … |
| maximal quotient of | #98860, #98862, #98884, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1648);