Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | Aut(G) = C213 (order 8192) · Inn(G) = C24 · Out(G) = C29 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C24 |
| commutator, socle, prime radical | C23 | C23×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
308 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,6)(7,8)(9,10)(11,14)(12,16)(13,15)(17,18,20,24,21,23,27,30)(19,22,26,29,25,28,31,32);(1,2)(3,7)(4,8)(5,6)(9,11,13,12)(10,14,15,16)(19,25)(22,28)(26,31)(29,32);(1,3,5,4)(2,7,6,8)(9,12,13,11)(10,14,15,16)(17,19)(18,22)(20,26)(21,25)(23,28)(24,29)(27,31)(30,32);(1,4,5,3)(2,7,6,8)(18,23)(19,25)(24,30)(26,31)Polycyclic presentation:
2^8|a^2=g,b^2=f,c^2=ef,d^2=e,g^2=h,b^a=be,c^a=cf,d^a=dh,c^b=ch,d^b=de,d^c=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#22834, #6380725, #6380727, … |
| maximal normal subgroup of | C2×#22834, #6380725, #6380727, … |
| maximal quotient of | #356389, #356432, #356435, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15822);