Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
529 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6)(7,11)(8,10)(9,12)(13,14)(15,22)(16,18)(17,19)(20,26)(21,25)(23,28)(24,27);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(14,18)(19,21)(20,27)(22,28);(2,4)(14,19)(17,25)(18,21)(20,28)(22,27)(24,26);(6,10)(7,12)(13,15,16,23)(14,20,18,27)(17,26,25,24)(19,28,21,22)Polycyclic presentation:
2^8|d^2=h,f^2=h,b^a=be,c^a=cf,d^a=def,f^a=fh,c^b=cg,d^b=deh,f^c=fh,f^d=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #6632026, #6632033, #6632042, … |
| maximal normal subgroup of | #6632026, #6632033, #6632042, … |
| maximal quotient of | #331792, #331793, #331796, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23401);