Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | Aut(G) order 393216 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, prime radical, cosocle | C23 | C25 |
| commutator | C22 | C24×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
900 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,4,3)(5,6)(7,13)(8,10)(9,16)(11,14)(12,17)(15,18)(19,20)(21,22,25,28)(23,29,31,26)(24,27,32,34)(30,35,36,33);(6,8)(7,14)(12,16)(15,20)(21,22,25,28)(23,26,31,29)(24,27,32,34)(30,33,36,35);(1,2,4,3)(5,7)(6,11)(8,13)(9,15)(10,14)(12,18)(16,19)(17,20)(21,22)(23,26)(24,27)(25,28)(29,31)(30,33)(32,34)(35,36);(5,8,10,6)(7,13,14,11)(9,16,17,12)(15,19,20,18)(21,23)(22,26)(24,30)(25,31)(27,33)(28,29)(32,36)(34,35);(1,3,4,2)(5,9)(6,12)(7,15)(8,16)(10,17)(11,18)(13,19)(14,20)(21,24,25,32)(22,27,28,34)(23,30,31,36)(26,33,29,35)Polycyclic presentation:
2^8|a^2=h,b^2=fg,c^2=fgh,d^2=f,e^2=h,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=df76 conjugacy classes — view the full character table.
| Fitting subgroup of | #34526⋊C3 |
| maximal subgroup of | #97047, #209190, #209191, … |
| maximal normal subgroup of | #97047, #209190, #209191, … |
| maximal quotient of | #513875, #513951, #528206, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27514);