Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| 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 | |
370 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,27,23,28,21,26)(19,24,29,25,20,30,31,32);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(17,19,21,31,23,20,22,29)(18,24,26,32,28,30,27,25);(1,3,4,7)(2,6,5,8)(10,14)(13,16)(17,20,23,19)(18,25,28,32)(21,31,22,29)(24,26,30,27);(2,5)(6,8)(17,21,23,22)(18,26,28,27)(19,29,20,31)(24,25,30,32)Polycyclic presentation:
2^8|a^2=g,b^2=fh,c^2=eh,d^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=dh,f^c=fh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#27710, #6457072, #6457087, … |
| maximal normal subgroup of | C2×#27710, #6457072, #6457087, … |
| maximal quotient of | #360358, #361369, #366380, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 20698);