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) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
328 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,6)(3,7,4,8)(9,10,13,16)(11,15,12,14)(17,18,20,23)(19,25,27,22)(21,24,29,31)(26,30,28,32);(2,6)(7,8)(9,11,13,12)(10,14,16,15)(17,19)(18,22)(20,27)(21,26)(23,25)(24,32)(28,29)(30,31);(1,3,5,4)(2,7,6,8)(9,12,13,11)(10,15,16,14)(18,23)(19,26)(21,29)(22,30)(25,32)(27,28);(1,4,5,3)(2,7,6,8)(10,16)(14,15)(18,23)(22,25)(24,31)(30,32)Polycyclic presentation:
2^8|a^2=e,b^2=fh,c^2=eh,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×#27492, #6451197, #6451215, … |
| maximal normal subgroup of | C2×#27492, #6451197, #6451215, … |
| maximal quotient of | #361527, #363611, #366255, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 20480);