Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2021 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #120 |
| commutator, prime radical | C24 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | #385 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
301 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 301 conjugacy classes of subgroups — too many to draw. Only its 111 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:
(5,6,9,14,10,13,22,28)(7,11,17,25,16,24,20,29)(8,12,19,27,18,26,21,30)(15,23,32,34,31,35,33,36);(1,2)(3,4)(5,7)(6,11,13,24)(8,15,18,31)(9,20)(10,16)(12,23)(14,29,28,25)(17,22)(19,33,21,32)(26,35)(27,36)(30,34);(2,4)(5,8)(6,12,13,26)(7,15,16,31)(9,21)(10,18)(11,23)(14,30,28,27)(17,33,20,32)(19,22)(24,35)(25,36)(29,34)Polycyclic presentation:
2^8|a^2=g,b^2=e,c^2=d,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=cf,d^b=dh,f^b=fh,g^b=gh,e^c=eh,f^c=fh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #93944, #136162, #136163, … |
| maximal normal subgroup of | #93944, #136162, #136163, … |
| maximal quotient of | #120618, #120621, #120622, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4276);