Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2192 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #447 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C2×#41 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
200 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 200 conjugacy classes of subgroups — too many to draw. Only its 75 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 — 3 generators on 36 points:
(1,2,4,3)(5,6,11,17)(7,19,8,22)(9,25,28,34)(10,18,12,16)(13,23,32,20)(14,21,33,24)(15,27,30,29)(26,35,36,31);(1,3,4,2)(5,7,11,8)(6,13,17,32)(9,26,28,36)(10,21,12,24)(14,30,33,15)(16,31,18,35)(19,34,22,25)(20,27,23,29);(5,8,11,7)(6,14,17,33)(9,23,28,20)(10,21,12,24)(13,30,32,15)(16,19,18,22)(25,31,34,35)(26,29,36,27)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=h,d^2=gh,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gh,d^b=dg,g^b=gh,e^c=eh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#12926, #494255, #494256, … |
| maximal normal subgroup of | C2×#12926, #494255, #494256, … |
| maximal quotient of | #135821, #135844, #135987, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5914);