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) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
384 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6,8,10)(7,9,11,12)(13,14,17,21)(15,22,18,19)(16,24,25,27)(20,23,28,26);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,15,17,18)(14,19,21,22)(16,23,25,26)(20,27,28,24);(2,4)(5,7,8,11)(6,9,10,12)(14,19)(15,18)(20,27)(21,22)(23,26)(24,28);(6,10)(9,12)(13,16,17,25)(14,20,21,28)(15,23,18,26)(19,27,22,24)Polycyclic presentation:
2^8|a^2=e,b^2=h,c^2=eh,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 | #6633079, #6633086, #6633093, … |
| maximal normal subgroup of | #6633079, #6633086, #6633093, … |
| maximal quotient of | #336908, #336910, #336916, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23672);