Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33359 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×#99 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C2×D4 | C24 |
| prime radical | #3500 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×D4 ◁ C2×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
626 conjugacy classes of subgroups; only the 181 normal subgroups are shown.
This group has 626 conjugacy classes of subgroups — too many to draw. Only its 181 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 — 4 generators on 32 points:
(1,2)(3,11,14,24)(4,8)(5,17,19,9)(6,10)(7,13)(12,25,26,32)(15,22)(16,29,28,23)(18,21)(20,27)(30,31);(1,3,6,14)(2,7,10,21)(4,12,15,26)(5,18,19,13)(8,20,22,31)(9,24,17,11)(16,30,28,27)(23,32,29,25);(1,4)(2,8)(3,12)(5,16)(6,15)(7,20)(9,23)(10,22)(11,25)(13,27)(14,26)(17,29)(18,30)(19,28)(21,31)(24,32);(4,15)(8,22)(12,26)(16,28)(20,31)(23,29)(25,32)(27,30)Polycyclic presentation:
2^8|a^2=f,b^2=h,e^2=h,b^a=be,e^a=eg,g^a=gh,e^b=eh,f^b=fgh,d^c=dh,f^e=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #151821, #151828, #152708, … |
| maximal normal subgroup of | #151821, #151828, #152708, … |
| maximal quotient of | #480016, #480046, #480117, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26349);