Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
400 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,4,3)(5,6)(7,12)(8,10)(9,11)(13,14,19,23)(15,20,16,25)(17,22,18,27)(21,24,28,26);(5,7,9,8)(6,10,11,12)(13,15,19,16)(14,20,23,25)(17,26,18,24)(21,27,28,22);(13,16,19,15)(14,21,23,28)(17,26,18,24)(20,27,25,22);(1,3,4,2)(5,8,9,7)(6,10,11,12)(13,17,19,18)(14,22,23,27)(15,24,16,26)(20,28,25,21)Polycyclic presentation:
2^8|a^2=g,b^2=eh,c^2=h,d^2=eg,f^2=h,b^a=be,c^a=cf,f^a=fh,d^b=dh,f^c=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#21625, #6373168, #6373174, … |
| maximal normal subgroup of | C2×#21625, #6373168, #6373174, … |
| maximal quotient of | #212330, #212338, #212370, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14613);