Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #3415 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D8 |
| commutator | C8 | C23×C4 |
| Frattini, cosocle | C2×C8 | C24 |
| prime radical | C22×D16 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #117 ◁ #405 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
359 conjugacy classes of subgroups; only the 159 normal subgroups are shown.
This group has 359 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) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6)(7,12)(8,11)(9,22)(10,13)(14,19)(15,20)(16,28)(17,23)(18,32)(21,24)(25,33)(26,31)(27,30)(29,35)(34,36);(6,11)(8,19)(9,20)(12,23)(13,24)(14,15)(17,31)(18,30)(21,33)(22,25)(26,27)(28,35)(29,36)(32,34);(1,3,4,2)(5,7)(6,12)(8,17)(9,18)(10,16)(11,23)(13,28)(14,26)(15,27)(19,31)(20,30)(21,29)(22,32)(24,35)(25,34)(33,36);(6,13)(7,16)(11,24)(14,25)(15,22)(17,29)(18,30)(31,36)Polycyclic presentation:
2^8|a^2=f,c^2=f,e^2=gh,g^2=h,b^a=be,d^a=dh,e^a=eg,g^a=gh,e^b=eg,g^b=gh,d^c=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #150149, #152284, #152438, … |
| maximal normal subgroup of | #150149, #152284, #152438, … |
| maximal quotient of | #483504, #483792, #483828, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26444);