Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | Aut(G) order 49152 |
| homology | Abelianization (H1) = C22×C4×C8 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C42 | C24 |
| commutator | C2 | C22×C4×C8 |
| socle | C22 | C22×C42 |
| prime radical | C2×#41 | C2×C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
445 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(9,10,12,16,13,14,19,21)(11,15,18,22,17,20,23,24);(1,2,3,5,4,6,7,8)(10,14)(15,20)(16,21)(22,24);(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24);(11,17)(15,20)(18,23)(22,24)Polycyclic presentation:
2^8|a^2=e,b^2=f,e^2=g,f^2=h,b^a=bg,d^c=dg136 conjugacy classes — view the full character table.
| Fitting subgroup of | #17316⋊C3 |
| maximal subgroup of | #126873, #128262, #137627, … |
| maximal normal subgroup of | #126873, #128262, #137627, … |
| maximal quotient of | #180790, #180849, #181806, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10304);