Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11115 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #388 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C2 | #1443 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
274 conjugacy classes of subgroups; only the 101 normal subgroups are shown.
This group has 274 conjugacy classes of subgroups — too many to draw. Only its 101 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,6,11,7,10,5,12)(3,13,15,24,17,8,16,25)(4,18,21,27,22,28,20,9)(14,26,19,29,30,32,31,23);(1,3,5,15,7,17,6,16)(2,8,12,25,10,13,11,24)(4,19,20,30,22,31,21,14)(9,26,28,29,27,32,18,23);(1,4)(2,9,10,27)(3,14)(5,20)(6,21)(7,22)(8,23,13,29)(11,18,12,28)(15,19)(16,31)(17,30)(24,32,25,26)Polycyclic presentation:
2^8|a^2=g,b^2=de,c^2=d,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=cf,f^b=fh,g^b=gh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11314, #358422, #360367, … |
| maximal normal subgroup of | C2×#11314, #358422, #360367, … |
| maximal quotient of | #121082, #121109, #121111, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4302);