Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
380 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,7)(4,5)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18,20,22)(19,25,27,23)(21,24,28,30)(26,31,29,32);(2,5)(6,7)(9,11,12,16)(10,13,14,15)(18,22)(19,26)(21,28)(23,31)(25,32)(27,29);(10,14)(13,15)(17,19,20,27)(18,23,22,25)(21,29,28,26)(24,31,30,32);(1,3,4,8)(2,6,5,7)(10,14)(13,15)(18,22)(23,25)(24,30)(31,32)Polycyclic presentation:
2^8|a^2=h,b^2=fh,c^2=h,d^2=ef,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#27404, #6449352, #6449371, … |
| maximal normal subgroup of | C2×#27404, #6449352, #6449371, … |
| maximal quotient of | #361419, #363585, #366093, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 20392);