Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2069 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #396 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C24 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
411 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 411 conjugacy classes of subgroups — too many to draw. Only its 119 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:
(2,4)(5,6,10,14,11,15,9,26)(7,16,20,28,19,32,18,12)(8,21,24,30,25,13,23,29)(17,33,35,36,22,31,34,27);(1,2,3,4)(5,7,9,18,11,19,10,20)(6,12,14,16,15,28,26,32)(8,22,23,35,25,17,24,34)(13,27,29,33,21,36,30,31);(5,8,11,25)(6,13,15,21)(7,17,19,22)(9,24,10,23)(12,27,28,36)(14,30,26,29)(16,31,32,33)(18,35,20,34)Polycyclic presentation:
2^8|a^2=g,b^2=d,c^2=h,d^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,c^b=cf,f^b=fh,d^c=dh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11140, #357835, C4×#2070, … |
| maximal normal subgroup of | C2×#11140, #357835, C4×#2070, … |
| maximal quotient of | #116994, #117028, #117030, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4128);