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 |
| prime radical | C2×2−1+4 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
3369 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,4,6)(3,7,8,5)(9,10,14,15)(11,13,17,22)(12,16,21,23)(18,20,19,24);(1,3,4,8)(2,5,6,7);(10,15)(13,22)(16,23)(20,24);(9,11,14,17)(10,13,15,22)(12,19,21,18)(16,24,23,20);(9,12,14,21)(10,16,15,23)(11,18,17,19)(13,20,22,24);(9,13)(10,17)(11,15)(12,20)(14,22)(16,19)(18,23)(21,24)Polycyclic presentation:
2^8|a^2=gh,b^2=g,d^2=h,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 | #63004⋊C3, #63004⋊C5 |
| maximal subgroup of | #347236, #347260, #347284, … |
| maximal normal subgroup of | #347236, #347260, #347284, … |
| maximal quotient of | C4⋊C4×2−1+4, #7625790, #7625817, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55992);