Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | Aut(G) = C213 (order 8192) · Inn(G) = C24 · Out(G) = C29 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C24 |
| commutator, socle | C23 | C23×C4 |
| prime radical | #392 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
441 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)(6,9)(7,11)(13,14,18,20,17,21,16,24)(15,19,22,28,23,27,25,26);(5,6,8,9)(7,10,11,12)(13,14,16,24,17,21,18,20)(15,19,25,26,23,27,22,28);(2,4)(6,9)(10,12)(13,15)(14,19)(16,25)(17,23)(18,22)(20,28)(21,27)(24,26);(5,7,8,11)(6,10,9,12)(13,16,17,18)(14,20,21,24)(15,22,23,25)(19,26,27,28)Polycyclic presentation:
2^8|a^2=g,b^2=eg,d^2=e,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=ceh,d^b=dh,d^c=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#22144, #6375810, #6375812, … |
| maximal normal subgroup of | C2×#22144, #6375810, #6375812, … |
| maximal quotient of | #218684, #218685, #218686, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15132);