Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C25 |
| commutator, Frattini, socle, cosocle | C22 | C26 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
3615 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,19)(13,16)(15,20)(17,23)(18,22)(21,24);(2,5)(6,8)(9,11,13,18)(10,14,16,22)(12,17,20,24)(15,21,19,23);(9,12)(10,15)(11,17)(13,20)(14,21)(16,19)(18,24)(22,23);(12,20)(15,19)(17,24)(21,23);(10,16)(14,22)(15,19)(21,23);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^8|b^2=h,f^2=g,b^a=bg,c^a=ch,e^a=eh,d^c=dh88 conjugacy classes — view the full character table.
| maximal subgroup of | #184920, #184922, #184924, … |
| maximal normal subgroup of | #184920, #184922, #184924, … |
| maximal quotient of | #7625424, #7625453, #7625490, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55769);