Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C16 | C23 |
| commutator, socle | C22 | C23×C8 |
| Frattini, cosocle | C2×C8 | C24 |
| prime radical | #532 | C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C16 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
345 conjugacy classes of subgroups; only the 203 normal subgroups are shown.
This group has 345 conjugacy classes of subgroups — too many to draw. Only its 203 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)(5,6,10,15,11,16,25,32,9,14,23,30,24,31,27,36)(7,17,19,33,20,21,29,35,18,12,26,28,34,8,13,22);(2,4)(5,7)(6,12)(8,16)(9,18)(10,19)(11,20)(13,27)(14,17)(15,28)(21,31)(22,32)(23,26)(24,34)(25,29)(30,33)(35,36);(5,8,10,22,11,17,25,33,9,21,23,35,24,12,27,28)(6,13,15,7,16,19,32,20,14,29,30,18,31,26,36,34);(6,14)(7,18)(13,29)(15,30)(16,31)(19,26)(20,34)(32,36)Polycyclic presentation:
2^8|a^2=f,c^2=f,f^2=g,g^2=h,b^a=be,d^a=dh,d^b=dh112 conjugacy classes — view the full character table.
| maximal subgroup of | #128463, #128484, #128504, … |
| maximal normal subgroup of | #128463, #128484, #128504, … |
| maximal quotient of | #483489, #483524, #483535, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26422);