Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2613 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C24 |
| commutator, socle, prime radical | C23 | C23×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
347 conjugacy classes of subgroups; only the 163 normal subgroups are shown.
This group has 347 conjugacy classes of subgroups — too many to draw. Only its 163 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) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,14)(8,12)(9,18)(10,13)(11,16)(15,20)(17,19)(21,22,25,29,26,23,27,31)(24,28,33,35,32,30,34,36);(5,7)(6,11)(8,17)(9,15)(10,16)(12,20)(13,14)(18,19)(21,23,25,31,26,22,27,29)(24,28,33,35,32,30,34,36);(1,2,3,4)(5,8,10,9)(6,12,13,18)(7,15,16,17)(11,19,14,20)(21,24)(22,28)(23,30)(25,33)(26,32)(27,34)(29,35)(31,36);(5,9,10,8)(6,12,13,18)(7,15,16,17)(11,20,14,19)(24,32)(28,30)(33,34)(35,36)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=ef,d^2=e,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=ceh,d^c=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #122421, #122429, #128969, … |
| maximal normal subgroup of | #122421, #122429, #128969, … |
| maximal quotient of | #219281, #219282, #219283, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15229);