Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | Aut(G) order 524288 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C25 |
| commutator | C22 | C24×C4 |
| prime radical | C24 | C24 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
955 conjugacy classes of subgroups; only the 703 normal subgroups are shown.
This group has 955 conjugacy classes of subgroups — too many to draw. Only its 703 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,3,4)(5,6,10,8)(7,11,15,13)(9,16,17,12)(14,19,20,18)(22,26)(23,29)(24,32)(34,36);(6,8)(11,13)(12,16)(18,19)(21,22,25,26)(23,30,29,27)(24,28,32,33)(31,36,35,34);(5,7,10,15)(6,11,8,13)(9,14,17,20)(12,18,16,19)(21,22)(23,27)(24,28)(25,26)(29,30)(31,34)(32,33)(35,36);(5,8,10,6)(7,13,15,11)(9,16,17,12)(14,19,20,18)(21,23,25,29)(22,27,26,30)(24,31,32,35)(28,34,33,36);(5,9)(6,12)(7,14)(8,16)(10,17)(11,18)(13,19)(15,20)(21,24)(22,28)(23,31)(25,32)(26,33)(27,34)(29,35)(30,36)Polycyclic presentation:
2^8|a^2=h,b^2=g,c^2=fg,d^2=f,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.
| maximal subgroup of | #107231, #107761, #108011, … |
| maximal normal subgroup of | #107231, #107761, #108011, … |
| maximal quotient of | #513785, #513811, #513835, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27360);