Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #117 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | #380 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
373 conjugacy classes of subgroups; only the 175 normal subgroups are shown.
This group has 373 conjugacy classes of subgroups — too many to draw. Only its 175 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:
(5,6)(7,15,18,11)(8,19)(9,22)(10,13)(12,20)(14,23)(16,29,31,33)(17,25,32,28)(21,34)(24,30,27,36)(26,35);(1,2,3,4)(5,7)(6,11)(8,16)(9,17)(10,18)(12,24)(13,15)(14,25)(19,33)(20,30)(21,31)(22,28)(23,32)(26,27)(29,34)(35,36);(5,8,9,20,10,21,23,35)(6,12,14,19,13,26,22,34)(7,16,17,30,18,31,32,36)(11,24,25,33,15,27,28,29)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=e,e^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^b=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | #93575, #93581, #134422, … |
| maximal normal subgroup of | #93575, #93581, #134422, … |
| maximal quotient of | #108859, #108860, #108861, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3854);