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, prime radical | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
336 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,7)(3,6,4,8)(10,13)(11,15)(17,18,21,24)(19,22,27,31)(20,23,28,32)(25,29,26,30);(1,2)(3,6)(4,8)(5,7)(9,10)(11,14)(12,13)(15,16)(17,19,21,27)(18,22,24,31)(20,25,28,26)(23,29,32,30);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,14,13,16)(19,25)(20,28)(22,29)(23,32)(26,27)(30,31);(1,4,5,3)(2,6,7,8)(10,13)(14,16)Polycyclic presentation:
2^8|a^2=efh,b^2=h,c^2=e,d^2=ef,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 | #6647795, #6647799, #6647823, … |
| maximal normal subgroup of | #6647795, #6647799, #6647823, … |
| maximal quotient of | #384910, #385020, #385118, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25596);