Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C8 | C24 |
| commutator, socle | C22 | C23×C8 |
| prime radical | #532 | C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
337 conjugacy classes of subgroups; only the 199 normal subgroups are shown.
This group has 337 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) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,9,15,10,16,24,32,11,14,25,30,26,31,27,36)(7,12,18,28,19,8,20,21,17,22,33,35,34,23,13,29);(2,4)(5,7)(6,12)(8,16)(9,18)(10,19)(11,17)(13,27)(14,22)(15,28)(20,24)(21,32)(23,31)(25,33)(26,34)(29,36)(30,35);(5,8,9,21,10,22,24,35,11,23,25,29,26,12,27,28)(6,13,15,7,16,18,32,19,14,20,30,17,31,33,36,34);(6,14)(7,17)(13,20)(15,30)(16,31)(18,33)(19,34)(32,36)Polycyclic presentation:
2^8|a^2=f,c^2=f,f^2=g,g^2=h,b^a=be,c^a=ch,d^a=dh,d^b=dh88 conjugacy classes — view the full character table.
| maximal subgroup of | #128464, #128483, #128514, … |
| maximal normal subgroup of | #128464, #128483, #128514, … |
| maximal quotient of | #483490, #483525, #483536, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26423);