Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| automorphism | Aut(G) order 32768 |
| 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 | C2×#469 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
545 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)(13,14,18,22,19,16,17,21)(15,20,26,23,24,27,25,28);(5,7,8,12)(6,9,10,11)(13,15,18,26,19,24,17,25)(14,20,22,23,16,27,21,28);(2,4)(13,16,19,14)(15,23,24,28)(17,21,18,22)(20,25,27,26);(6,10)(9,11)(15,24)(20,27)(23,28)(25,26)Polycyclic presentation:
2^8|a^2=fh,b^2=eg,c^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=dh,f^c=fh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#28363, #6459731, #6459740, … |
| maximal normal subgroup of | C2×#28363, #6459731, #6459740, … |
| maximal quotient of | #319346, #319360, #319377, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21351);