Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
313 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,6)(3,7,8,5)(19,25)(20,26)(22,28)(24,29)(27,31)(30,32);(1,3,4,8)(2,5,6,7)(9,10)(11,14)(12,13)(15,16)(17,18,21,23)(19,22,27,30)(20,24,26,29)(25,28,31,32);(10,13)(14,16)(17,19,26,31,21,27,20,25)(18,22,29,32,23,30,24,28);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(18,23)(22,30)(24,29)(28,32)Polycyclic presentation:
2^8|a^2=e,b^2=eh,c^2=fh,d^2=g,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,d^b=dh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#29530, #6461767, #6461772, … |
| maximal normal subgroup of | C2×#29530, #6461767, #6461772, … |
| maximal quotient of | #363758, #364213, #371598, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22518);