Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17597 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×C22⋊C4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C23 | C22×D4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
437 conjugacy classes of subgroups; only the 183 normal subgroups are shown.
This group has 437 conjugacy classes of subgroups — too many to draw. Only its 183 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) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,9)(4,12)(5,7)(6,10)(8,14)(11,16)(13,15)(18,23,25,29)(24,30,32,31);(1,3,5,11)(2,6,7,15)(4,13,14,10)(8,16,12,9)(17,18)(19,24)(20,23)(21,29)(22,25)(26,31)(27,32)(28,30);(2,7)(6,15)(8,12)(9,16)(17,19,22,27)(18,24,25,32)(20,28,21,26)(23,30,29,31);(17,20,22,21)(18,23,25,29)(19,26,27,28)(24,31,32,30)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=gh,d^2=gh,e^2=h,b^a=be,c^a=cg,e^a=eg,e^b=eh,f^b=fgh,d^c=dgh49 conjugacy classes — view the full character table.
| maximal subgroup of | #221193, #221194, #221195, … |
| maximal normal subgroup of | #221193, #221194, #221195, … |
| maximal quotient of | #160640, #160641, #160902, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10620);