Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
799 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(5,6,8,7)(9,10)(11,19)(12,17)(13,16)(14,18)(15,21)(20,23)(22,24);(2,4)(5,7,8,6)(9,11)(10,15)(12,22)(13,21)(14,20)(16,23)(17,19)(18,24);(10,16)(12,13)(15,23)(17,18)(19,24)(21,22);(11,20)(15,24)(19,23)(21,22)Polycyclic presentation:
2^8|a^2=g,b^2=g,e^2=h,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^b=dh,e^b=eh,f^b=fh,e^c=eh,f^c=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | #137730, #137732, #137798, … |
| maximal normal subgroup of | #137730, #137732, #137798, … |
| maximal quotient of | #191866, #191881, #191896, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13864);