Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1064 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,9,7)(8,11,12,10)(13,14)(15,18)(16,24)(17,20)(19,25)(21,23)(22,27)(26,28);(6,7)(10,11)(13,14,17,20)(15,18,23,21)(16,24,25,19)(22,27,28,26);(1,2,3,4)(13,14)(15,21)(16,19)(17,20)(18,23)(22,27)(24,25)(26,28);(5,7,9,6)(8,11,12,10)(13,15)(14,18)(16,22)(17,23)(19,26)(20,21)(24,27)(25,28);(5,8)(6,10)(7,11)(9,12)(13,16,17,25)(14,19,20,24)(15,22,23,28)(18,26,21,27)Polycyclic presentation:
2^8|a^2=fh,b^2=h,c^2=g,d^2=fh,e^2=h,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfh,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #133589, #500793, #503218, … |
| maximal normal subgroup of | #133589, #500793, #503218, … |
| maximal quotient of | #560451, #561547, #562957, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 45073);