Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| 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×#120 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1015 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6,10,11)(7,12,14,9)(8,16,18,13)(15,17,20,19)(21,22,26,30)(23,27,31,36)(24,28,34,33)(25,29,35,32);(6,11)(7,14)(13,16)(15,20)(23,31)(25,35)(27,36)(29,32);(1,2,3,4)(5,7)(6,12)(8,15)(9,11)(10,14)(13,19)(16,17)(18,20)(21,23)(22,27)(24,29)(25,33)(26,31)(28,35)(30,36)(32,34);(5,8,10,18)(6,13,11,16)(7,15,14,20)(9,17,12,19)(21,24)(22,28)(23,32)(25,27)(26,34)(29,31)(30,33)(35,36);(5,9,10,12)(6,7,11,14)(8,17,18,19)(13,15,16,20)(21,25)(22,29)(23,33)(24,27)(26,35)(28,31)(30,32)(34,36)Polycyclic presentation:
2^8|a^2=h,c^2=g,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #482852, #6353183, #6353193, … |
| maximal normal subgroup of | #482852, #6353183, #6353193, … |
| maximal quotient of | #544539, #545092, #545539, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 34406);