Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62817 |
| 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 | |
3665 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)(3,4)(5,6)(7,8)(10,14)(12,19)(15,20)(18,23);(2,6)(4,7)(11,17)(15,20)(18,23)(22,24);(1,2,5,6)(3,7,8,4)(9,10,13,14)(11,15,17,20)(12,16,19,21)(18,22,23,24);(1,3)(2,7)(4,6)(5,8);(1,4,5,7)(2,3,6,8)(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24);(1,3)(2,7)(4,6)(5,8)(9,12)(10,16)(11,18)(13,19)(14,21)(15,22)(17,23)(20,24)Polycyclic presentation:
2^8|c^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=cg,e^b=eh76 conjugacy classes — view the full character table.
| maximal subgroup of | #185268, #185274, #185280, … |
| maximal normal subgroup of | #185268, #185274, #185280, … |
| maximal quotient of | #7626211, #7626224, #7626248, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55812);