Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30396 |
| 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 | #1999 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
506 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,3,4)(5,6)(7,11)(8,10)(9,12)(13,14)(15,21)(16,19)(17,20)(18,23)(22,28)(24,27)(25,26);(5,7)(6,9)(8,12)(10,11)(15,22)(17,24)(18,26)(20,27)(21,28)(23,25);(2,4)(13,15,16,23)(14,18,19,21)(17,25,24,22)(20,28,27,26);(2,4)(6,10)(7,12)(14,19)(18,21)(20,27)(26,28)Polycyclic presentation:
2^8|a^2=fh,c^2=h,g^2=h,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,g^b=gh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #6632124, #6632131, #6632137, … |
| maximal normal subgroup of | #6632124, #6632131, #6632137, … |
| maximal quotient of | #342115, #342116, #342119, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23982);