Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2190 |
| automorphism | Aut(G) = #72282918 (order 2048) · Inn(G) = #447 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #447 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #1879 |
| prime radical | #3414 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
233 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 233 conjugacy classes of subgroups — too many to draw. Only its 67 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:
(1,2,7,13,8,14,6,25,9,15,26,32,24,31,5,12)(3,16,17,29,18,30,4,20,19,11,22,10,21,27,23,28);(1,3,8,21,9,19,24,18)(2,10,14,20,15,29,31,28)(4,6,17,26,23,5,22,7)(11,12,30,13,16,25,27,32);(1,4)(2,11)(3,6)(5,19)(7,18)(8,17)(9,23)(10,12)(13,20)(14,30)(15,16)(21,26)(22,24)(25,29)(27,31)(28,32)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,d^2=gh,e^2=gh,f^2=g,g^2=h,b^a=bd,c^a=ce,e^a=eh,d^b=dg,e^b=egh,f^b=fg,g^b=gh,d^c=dgh,e^c=eg,f^c=fgh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13024, #494531, #494999, … |
| maximal normal subgroup of | C2×#13024, #494531, #494999, … |
| maximal quotient of | #136695, #136696, #136729, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6012);