Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2192 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #447 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C2×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
201 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 201 conjugacy classes of subgroups — too many to draw. Only its 69 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 — 3 generators on 36 points:
(1,2,4,3)(5,6)(7,18)(8,12)(9,27)(10,28)(11,17)(13,20)(14,23)(15,26)(16,25)(19,32)(21,31)(22,34)(24,33)(29,30)(35,36);(1,3,4,2)(6,12)(8,23)(9,26)(10,25)(13,30)(14,16)(15,17)(18,32)(19,20)(21,29)(24,36)(27,28)(31,33)(34,35);(5,7)(6,13)(8,24)(9,19)(10,29)(11,22)(12,30)(14,34)(15,31)(16,35)(17,33)(18,27)(20,26)(21,25)(23,36)(28,32)Polycyclic presentation:
2^8|a^2=f,b^2=f,d^2=gh,e^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=egh,g^a=gh,d^b=dg,e^b=egh,g^b=gh,d^c=dgh,e^c=egh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#12945, #494313, #494314, … |
| maximal normal subgroup of | C2×#12945, #494313, #494314, … |
| maximal quotient of | #135888, #135926, #136147, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5933);