Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| 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 |
| prime radical | C22×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
546 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6)(7,11)(8,10)(9,12)(13,14,19,22)(15,17,21,18)(16,23,26,27)(20,25,28,24);(5,7,8,12)(6,9,10,11)(13,15)(14,17)(16,20)(18,22)(19,21)(23,25)(24,27)(26,28);(1,2,3,4)(13,16,18,25,19,26,17,24)(14,20,21,27,22,28,15,23);(6,10)(9,11)(13,17,19,18)(14,21,22,15)(16,24,26,25)(20,27,28,23)Polycyclic presentation:
2^8|a^2=h,b^2=e,c^2=fh,d^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,c^b=cg,f^b=fh,g^b=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#25542, #6452130, #6452149, … |
| maximal normal subgroup of | C2×#25542, #6452130, #6452149, … |
| maximal quotient of | #290970, #290971, #290974, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18530);