Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3198 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
337 conjugacy classes of subgroups; only the 159 normal subgroups are shown.
This group has 337 conjugacy classes of subgroups — too many to draw. Only its 159 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:
(1,2,4,3)(5,6)(7,14)(8,12)(9,18)(10,13)(11,16)(15,20)(17,19)(21,22)(23,30)(24,28)(25,34)(26,29)(27,32)(31,35)(33,36);(1,3,4,2)(5,7)(6,11)(8,17)(9,15)(10,16)(12,20)(13,14)(18,19)(21,22)(23,27)(24,34)(25,28)(26,29)(30,32)(31,36)(33,35);(5,8,10,9)(6,12,13,18)(7,15,16,17)(11,19,14,20)(21,23,25,32,26,33,24,31)(22,27,28,30,29,35,34,36);(5,9,10,8)(6,12,13,18)(7,15,16,17)(11,20,14,19)(21,24,26,25)(22,28,29,34)(23,31,33,32)(27,30,35,36)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=ef,d^2=eh,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce,d^b=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #219607, #219612, #219621, … |
| maximal normal subgroup of | #219607, #219612, #219621, … |
| maximal quotient of | #219265, #219269, #219291, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15241);