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 |
| prime radical | C2×H(3,𝔽4) | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1116 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,9,10)(7,11,12,8)(13,14)(15,22)(16,20)(17,25)(18,19)(21,26)(23,24)(27,28);(6,10)(8,11)(14,19)(15,20)(17,26)(24,28);(1,2,3,4)(5,7)(6,8)(9,12)(10,11)(13,15)(14,16)(17,23)(18,20)(19,22)(21,24)(25,28)(26,27);(5,8)(6,7)(9,11)(10,12)(13,16)(14,20)(15,19)(17,24)(18,22)(21,27)(23,25)(26,28);(5,7)(6,8)(9,12)(10,11)(13,17)(14,21)(15,23)(16,24)(18,26)(19,25)(20,27)(22,28)Polycyclic presentation:
2^8|a^2=fh,c^2=fg,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 | #133580, #500826, #503241, … |
| maximal normal subgroup of | #133580, #500826, #503241, … |
| maximal quotient of | #560416, #561518, #562800, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 41800);