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) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#127 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
511 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,3)(5,6,8,7)(9,10)(11,18)(12,22)(13,16)(14,17)(15,21)(19,23)(20,24);(1,2,4,3)(5,7,8,6)(9,11)(10,15)(12,20)(13,21)(14,19)(16,18)(17,24)(22,23);(6,7)(10,16)(11,19)(12,14)(15,23)(18,24)(20,21);(1,3,4,2)(9,12,13,14)(10,17,16,22)(11,20,21,19)(15,24,18,23)Polycyclic presentation:
2^8|a^2=fh,b^2=ef,d^2=e,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×#25377, #6444152, #6444161, … |
| maximal normal subgroup of | C2×#25377, #6444152, #6444161, … |
| maximal quotient of | #287827, #287828, #287829, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18365);