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×#127 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
996 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)(5,6,10,14)(7,11,15,20)(8,12,18,17)(9,13,19,16)(21,22)(23,29)(24,27)(25,28)(26,31)(30,35)(32,34)(33,36);(7,15)(9,19)(11,20)(13,16)(21,22,25,28)(23,26,31,29)(24,27,32,34)(30,33,36,35);(1,2,3,4)(5,7)(6,11)(8,13)(9,17)(10,15)(12,19)(14,20)(16,18)(21,22)(23,26)(24,27)(25,28)(29,31)(30,33)(32,34)(35,36);(5,8)(6,12)(7,16)(9,11)(10,18)(13,15)(14,17)(19,20)(21,23,25,31)(22,26,28,29)(24,30,32,36)(27,33,34,35);(5,9)(6,13)(7,17)(8,11)(10,19)(12,15)(14,16)(18,20)(21,24,25,32)(22,27,28,34)(23,30,31,36)(26,33,29,35)Polycyclic presentation:
2^8|a^2=fh,b^2=f,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 | #482833, #6353264, #6353274, … |
| maximal normal subgroup of | #482833, #6353264, #6353274, … |
| maximal quotient of | #544545, #545092, #545544, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 34409);