Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30396 |
| 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 | C22×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
349 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,7)(3,6,4,8)(10,13)(11,15)(17,18,20,23)(19,25,27,22)(21,24,29,32)(26,31,28,30);(1,2)(3,6)(4,8)(5,7)(9,10)(11,14)(12,13)(15,16)(17,19,20,27)(18,22,23,25)(21,26,29,28)(24,30,32,31);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,14,13,16)(19,26)(21,29)(22,30)(24,32)(25,31)(27,28);(1,4,5,3)(2,6,7,8)(10,13)(14,16)(18,23)(19,27)(24,32)(26,28)Polycyclic presentation:
2^8|a^2=ef,b^2=h,c^2=eh,d^2=efh,g^2=h,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,g^b=gh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #6647793, #6647797, #6647821, … |
| maximal normal subgroup of | #6647793, #6647797, #6647821, … |
| maximal quotient of | #384913, #385024, #385119, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25591);