Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23387 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
437 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6)(7,12)(8,10)(9,11)(13,14,16,21,17,20,24,19)(15,18,23,26,22,27,28,25);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15)(14,18)(16,23)(17,22)(19,25)(20,27)(21,26)(24,28);(2,4)(5,8,9,7)(6,10,11,12)(14,19)(16,24)(18,25)(20,21)(23,28)(26,27);(14,20)(15,22)(19,21)(23,28)Polycyclic presentation:
2^8|a^2=f,b^2=e,c^2=e,f^2=h,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=dh,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #396675, #396676, #396677, … |
| maximal normal subgroup of | #396675, #396676, #396677, … |
| maximal quotient of | #243218, #243219, #243220, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16659);