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 | |
3693 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 6 generators on 24 points:
(2,5)(6,8)(9,10)(11,16)(12,22)(13,23)(14,17)(15,21)(18,24)(19,20);(9,11)(10,15)(12,19)(13,20)(14,21)(16,17)(18,23)(22,24);(1,2)(3,6)(4,5)(7,8)(9,12)(10,13)(11,18)(14,23)(15,24)(16,20)(17,22)(19,21);(3,7)(6,8)(9,12)(10,13)(11,19)(14,23)(15,20)(16,24)(17,22)(18,21);(1,3,4,7)(2,6,5,8)(9,11)(10,16)(12,18)(13,20)(14,21)(15,17)(19,23)(22,24);(1,2)(3,6)(4,5)(7,8)(9,13)(10,12)(11,20)(14,22)(15,19)(16,18)(17,23)(21,24)Polycyclic presentation:
2^8|e^2=gh,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 | #347078, #347102, #347126, … |
| maximal normal subgroup of | #347078, #347102, #347126, … |
| maximal quotient of | #7626597, #7626657, #7626692, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55906);