Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of B(4,𝔽2) |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #120 |
| commutator | C23 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | #379 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
607 conjugacy classes of subgroups; only the 175 normal subgroups are shown.
This group has 607 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:
(5,6,9,14,10,13,22,27)(7,15,18,28,17,11,21,24)(8,12,20,25,19,26,32,36)(16,23,31,34,30,29,33,35);(1,2,3,4)(5,7)(6,11)(8,16)(9,21)(10,17)(12,23)(13,15)(14,28)(18,22)(19,30)(20,33)(24,27)(25,35)(26,29)(31,32)(34,36);(5,8)(6,12)(7,16)(9,20)(10,19)(11,23)(13,26)(14,25)(15,29)(17,30)(18,31)(21,33)(22,32)(24,34)(27,36)(28,35)Polycyclic presentation:
2^8|a^2=f,b^2=g,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^b=eh,f^b=fh,d^c=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #146603, #146604, #146615, … |
| maximal normal subgroup of | #146603, #146604, #146615, … |
| maximal quotient of | #106077, #106078, #106081, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3735);