Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29736 |
| 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 ◁ #376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
314 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,5)(3,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,18,21,24)(19,20,26,23)(22,25,30,28)(27,29,32,31);(2,5)(6,8)(17,19,22,27,21,26,30,32)(18,23,25,31,24,20,28,29);(1,3)(2,6)(4,7)(5,8)(10,13)(14,16)(17,20,21,23)(18,19,24,26)(22,31,30,29)(25,32,28,27);(3,7)(6,8)(9,11,12,15)(10,14,13,16)(18,24)(20,23)(25,28)(29,31)Polycyclic presentation:
2^8|a^2=e,b^2=g,c^2=h,d^2=fh,g^2=h,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#30351, #6434269, #6434278, … |
| maximal normal subgroup of | C2×#30351, #6434269, #6434278, … |
| maximal quotient of | #380889, #380892, #380919, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23339);