Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C22 | C26 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
3585 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(2,5)(6,8)(9,10)(11,15)(12,16)(13,14)(17,23)(18,21)(19,24)(20,22);(1,2)(3,6)(4,5)(7,8);(10,14)(15,21)(16,22)(23,24);(1,3,4,7)(2,6,5,8)(9,11,13,18)(10,15,14,21)(12,19,20,17)(16,24,22,23);(9,12,13,20)(10,16,14,22)(11,17,18,19)(15,23,21,24);(9,10)(11,15)(12,16)(13,14)(17,23)(18,21)(19,24)(20,22)Polycyclic presentation:
2^8|d^2=gh,e^2=h,b^a=bg,c^a=ch,f^c=fh,e^d=eh85 conjugacy classes — view the full character table.
| Fitting subgroup of | #63000⋊C3 |
| maximal subgroup of | #347244, #347268, #347292, … |
| maximal normal subgroup of | #347244, #347268, #347292, … |
| maximal quotient of | #7625787, #7625788, #7625818, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55988);