Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29902 |
| automorphism | Aut(G) = C2×#72236721 (order 4096) · Inn(G) = #2622 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2622 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×H(5,𝔽2) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
378 conjugacy classes of subgroups; only the 101 normal subgroups are shown.
This group has 378 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 — 4 generators on 32 points:
(1,2)(3,12)(4,17)(5,22)(6,10)(7,11)(8,14)(9,19)(13,30)(15,24)(16,29)(18,23)(20,26)(21,27)(25,31)(28,32);(1,3,5,15,7,14,6,16)(2,8,10,24,11,12,22,29)(4,18,19,13,21,32,20,31)(9,25,17,28,26,30,27,23);(1,4,7,21)(2,9,11,26)(3,13,14,31)(5,20,6,19)(8,23,12,28)(10,27,22,17)(15,18,16,32)(24,30,29,25);(1,2)(3,8)(4,9)(5,10)(6,22)(7,11)(12,14)(13,28)(15,24)(16,29)(17,19)(18,25)(20,27)(21,26)(23,31)(30,32)Polycyclic presentation:
2^8|b^2=f,c^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,f^a=fh,c^b=cg,e^b=eh,d^c=de,f^c=fh,g^c=gh,g^d=gh34 conjugacy classes — view the full character table.
| maximal subgroup of | #500251, #500252, #500256, … |
| maximal normal subgroup of | #500251, #500252, #500256, … |
| maximal quotient of | #330925, #330928, #330931, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22891);