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) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
421 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6,8,9)(7,11,12,10)(13,14)(15,20)(16,25)(17,22)(18,27)(19,23)(21,26)(24,28);(1,2)(3,4)(6,9)(7,12)(13,15)(14,20)(16,19)(17,26)(18,24)(21,22)(23,25)(27,28);(2,4)(13,16,19,15)(14,21,23,28)(17,26,18,24)(20,27,25,22);(5,7,8,12)(6,10,9,11)(13,17)(15,24)(16,26)(18,19)(20,25)(22,27)Polycyclic presentation:
2^8|a^2=e,c^2=h,d^2=e,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 | #6632674, #6632676, #6632682, … |
| maximal normal subgroup of | #6632674, #6632676, #6632682, … |
| maximal quotient of | #340687, #340689, #340695, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23912);