Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11508 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#99 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C22×Q8 | C23 |
| prime radical | C23 | C2×C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
216 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 216 conjugacy classes of subgroups — too many to draw. Only its 65 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 — 3 generators on 36 points:
(2,4)(5,6,10,15,11,16,8,20)(7,12,17,27,19,28,21,35)(9,14,22,30,24,31,23,13)(18,26,33,25,34,36,32,29);(1,2,3,4)(5,7)(6,12)(8,21)(9,18)(10,17)(11,19)(13,29)(14,26)(15,27)(16,28)(20,35)(22,33)(23,32)(24,34)(25,30)(31,36);(5,8,11,10)(6,13,16,30)(7,17,19,21)(9,23,24,22)(12,25,28,29)(14,15,31,20)(18,33,34,32)(26,35,36,27)Polycyclic presentation:
2^8|a^2=f,b^2=d,c^2=h,e^2=h,f^2=h,b^a=bd,c^a=ce,e^a=eg,g^a=gh,c^b=ch,e^c=eh,f^c=fgh,f^e=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #130277, #130279, #130339, … |
| maximal normal subgroup of | #130277, #130279, #130339, … |
| maximal quotient of | #134233, #134257, #134287, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5792);