Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #117 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C22 | #432 |
| prime radical | C24 | M16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
231 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 231 conjugacy classes of subgroups — too many to draw. Only its 99 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:
(1,2)(3,4)(5,6)(7,17,20,28)(8,14)(9,15)(10,16)(11,13)(12,19,25,18)(21,31,34,36)(22,29)(23,27)(24,30)(26,33,35,32);(2,4)(5,7,9,21,10,19,24,33,11,20,23,34,8,18,22,32)(6,12,15,26,16,17,30,31,13,25,27,35,14,28,29,36);(6,13)(7,18,20,19)(9,23)(12,17,25,28)(14,16)(21,33,34,32)(22,24)(26,36,35,31)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=e,d^2=h,f^2=g,g^2=h,b^a=bd,c^a=ch,c^b=cg,d^b=dh,e^b=eh,f^c=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | #93548, #93552, #93553, … |
| maximal normal subgroup of | #93548, #93552, #93553, … |
| maximal quotient of | #128261, #128280, #128355, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5596);