Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3186 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C2×C4×D4 |
| prime radical | C23 | C2×#41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
307 conjugacy classes of subgroups; only the 155 normal subgroups are shown.
This group has 307 conjugacy classes of subgroups — too many to draw. Only its 155 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) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,10)(4,8)(5,14)(6,9)(7,12)(11,16)(13,15)(17,18,22,26,23,27,21,31)(19,24,28,32,29,25,20,30);(1,3)(2,7)(4,13)(5,11)(6,12)(8,16)(9,10)(14,15)(17,19,22,28,23,29,21,20)(18,24,26,32,27,25,31,30);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,10,16)(17,20,21,29,23,28,22,19)(18,25,26,30,27,24,31,32);(1,5,6,4)(2,8,9,14)(3,11,12,13)(7,16,10,15)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=ef,d^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce64 conjugacy classes — view the full character table.
| maximal subgroup of | #137896, #139124, #139125, … |
| maximal normal subgroup of | #137896, #139124, #139125, … |
| maximal quotient of | #219252, #219253, #219254, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15244);