Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62862 |
| 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 | |
3333 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2)(3,11)(4,15)(5,9)(6,10)(7,14)(8,17)(12,27)(13,26)(16,30)(18,23)(19,25)(20,31)(21,29)(22,28)(24,32);(1,3,5,14)(2,7,9,11)(4,16,18,12)(6,13,19,29)(8,22,24,20)(10,21,25,26)(15,27,23,30)(17,31,32,28);(1,4,5,18)(2,8,9,24)(3,12,14,16)(6,17,19,32)(7,20,11,22)(10,15,25,23)(13,28,29,31)(21,30,26,27);(2,9)(4,17)(7,11)(8,23)(10,25)(12,28)(15,24)(16,31)(18,32)(20,27)(21,26)(22,30);(3,13)(7,21)(11,26)(12,28)(14,29)(16,31)(20,30)(22,27);(2,10)(7,21)(8,15)(9,25)(11,26)(20,30)(22,27)(23,24)Polycyclic presentation:
2^8|b^2=g,c^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=cg,e^b=eh,d^c=dh70 conjugacy classes — view the full character table.
| maximal subgroup of | #6352342, #6352346, #6352374, … |
| maximal normal subgroup of | #6352342, #6352346, #6352374, … |
| maximal quotient of | #7626348, #7626349, #7626387, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55852);