Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3290 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#135 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
423 conjugacy classes of subgroups; only the 199 normal subgroups are shown.
This group has 423 conjugacy classes of subgroups — too many to draw. Only its 199 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) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4)(5,6)(7,19)(8,23)(9,15)(10,17)(11,16)(12,18)(13,21)(14,20)(22,35)(24,32)(25,33)(26,31)(27,36)(28,30)(29,34);(5,7)(6,13)(8,9)(10,24)(11,21)(12,22)(14,30)(15,33)(16,35)(17,19)(18,32)(20,27)(23,36)(25,29)(26,28)(31,34);(5,8,10,25,12,26,11,20)(6,14,16,31,18,33,17,23)(7,9,24,29,22,28,21,27)(13,30,35,34,32,15,19,36);(5,9)(6,15)(7,20)(8,24)(10,29)(11,27)(12,28)(13,31)(14,19)(16,36)(17,34)(18,30)(21,26)(22,25)(23,32)(33,35)Polycyclic presentation:
2^8|a^2=g,c^2=fh,e^2=h,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^b=de,e^b=eh,e^d=eh58 conjugacy classes — view the full character table.
| maximal subgroup of | #136389, #136423, #136862, … |
| maximal normal subgroup of | #136389, #136423, #136862, … |
| maximal quotient of | #225496, #225501, #225506, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15679);