Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62892 |
| 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 | |
3714 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2)(3,7)(4,5)(6,8)(10,15)(11,17)(13,21)(20,23);(2,5)(6,7)(9,10,14,15)(11,12,17,16)(13,18,21,22)(19,20,24,23);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,15)(13,19)(14,17)(18,23)(20,22)(21,24);(1,3)(2,6)(4,8)(5,7)(9,11)(10,12)(13,19)(14,17)(15,16)(18,20)(21,24)(22,23);(9,12)(10,17)(11,15)(13,20)(14,16)(18,24)(19,22)(21,23);(9,13)(10,18)(11,19)(12,20)(14,21)(15,22)(16,23)(17,24)Polycyclic presentation:
2^8|b^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=cg,e^d=eh73 conjugacy classes — view the full character table.
| maximal subgroup of | #6352583, #6352613, #6352643, … |
| maximal normal subgroup of | #6352583, #6352613, #6352643, … |
| maximal quotient of | #7626578, #7626581, #7626663, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55888);