Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| unambiguous semidirect product | C4⋊#564 |
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
878 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6)(7,13)(8,16)(9,12)(10,15)(11,17)(14,18)(19,20)(21,22,26,24)(23,29,31,27)(25,32,33,28)(30,34,36,35);(5,6,9,12)(7,10,15,13)(8,16,17,11)(14,19,20,18)(21,23,26,31)(22,27,24,29)(25,30,33,36)(28,34,32,35);(1,2,3,4)(5,6)(7,10)(8,11)(9,12)(13,15)(14,18)(16,17)(19,20);(5,7)(6,10)(8,14)(9,15)(11,18)(12,13)(16,19)(17,20)(21,24,26,22)(23,27,31,29)(25,32,33,28)(30,34,36,35);(5,8,9,17)(6,11,12,16)(7,14,15,20)(10,18,13,19)(21,25,26,33)(22,28,24,32)(23,30,31,36)(27,34,29,35)Polycyclic presentation:
2^8|a^2=fh,b^2=f,c^2=g,d^2=fh,e^2=f,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=dfh,e^b=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #97195, #97345, #172177, … |
| maximal normal subgroup of | #97195, #97345, #172177, … |
| maximal quotient of | #557080, #557389, #557700, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 39136);