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 | |
3522 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(2,6)(7,8)(9,10,14,15)(11,16,21,24)(12,17,22,20)(13,18,23,19);(1,2)(3,7)(4,8)(5,6);(10,15)(13,23)(16,24)(17,20);(1,3,5,4)(2,7,6,8)(9,11)(10,16)(12,18)(13,20)(14,21)(15,24)(17,23)(19,22);(1,4,5,3)(2,8,6,7)(9,12)(10,17)(11,19)(13,16)(14,22)(15,20)(18,21)(23,24);(9,13)(10,18)(11,20)(12,16)(14,23)(15,19)(17,21)(22,24)Polycyclic presentation:
2^8|a^2=h,d^2=g,e^2=g,b^a=bg,c^a=ch,f^c=fh,e^d=eh85 conjugacy classes — view the full character table.
| Fitting subgroup of | #62993⋊C3 |
| maximal subgroup of | #6595655, #6595661, #6595667, … |
| maximal normal subgroup of | #6595655, #6595661, #6595667, … |
| maximal quotient of | #7625775, #7625781, #7625782, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55981);