Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7318 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1304 |
| commutator | C22×C4 | C42 |
| Frattini, prime radical, cosocle | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C23 ▷ C2 ▷ C1 | |
228 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,5,9,6,10,17,24)(3,11,13,25,14,21,27,31)(4,8,15,22,16,23,28,32)(7,18,19,26,20,29,30,12);(1,3)(2,7)(4,12)(5,13,17,27)(6,14)(8,21)(9,19,24,30)(10,20)(11,23)(15,18,28,29)(16,26)(22,31,32,25)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=h,d^2=h,b^a=bc,c^a=cf,e^a=egh,g^a=gh,c^b=cg,d^b=dfh,f^b=fh,e^d=eh28 conjugacy classes — view the full character table.
| maximal subgroup of | #94624, #94625, #94699, … |
| maximal normal subgroup of | #94624, #94625, #94699, … |
| maximal quotient of | #93122, #93123, #93124, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 307);