Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| 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 | #2431 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
511 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)(15,19)(16,18)(17,20)(21,28)(22,25)(23,26)(24,27);(5,7,8,12)(6,9,10,11)(14,18)(15,21)(17,23)(19,27)(22,24)(25,28);(2,4)(13,15)(14,19)(16,22)(17,24)(18,25)(20,27)(21,23)(26,28);(6,10)(9,11)(14,18)(15,22)(20,26)(21,24)Polycyclic presentation:
2^8|a^2=f,b^2=eh,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,d^c=dh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#24490, #6451910, #6451916, … |
| maximal normal subgroup of | C2×#24490, #6451910, #6451916, … |
| maximal quotient of | #274171, #274172, #274173, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17478);