Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| 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×#540 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1013 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6,10,7)(8,11,15,13)(9,14,17,12)(16,19,20,18)(21,22)(23,28)(24,29)(25,33)(26,27)(30,34)(31,32)(35,36);(6,7)(8,15)(9,17)(18,19)(22,27)(24,28)(30,33)(32,35);(1,2)(3,4)(5,7,10,6)(8,11,15,13)(9,14,17,12)(16,18,20,19)(21,23,26,29)(22,28,27,24)(25,31,34,36)(30,35,33,32);(5,8,10,15)(6,11,7,13)(9,16,17,20)(12,18,14,19)(21,24,26,28)(22,29,27,23)(25,32,34,35)(30,36,33,31);(5,9)(6,12)(7,14)(8,16)(10,17)(11,18)(13,19)(15,20)(21,25)(22,30)(23,31)(24,32)(26,34)(27,33)(28,35)(29,36)Polycyclic presentation:
2^8|a^2=h,c^2=f,d^2=f,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=df,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #133768, #246338, #246378, … |
| maximal normal subgroup of | #133768, #246338, #246378, … |
| maximal quotient of | #576622, #576672, #576707, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 39805);