Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #560 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C24 |
| commutator | C22 | C22×C42 |
| socle | C23 | C23×C4 |
| prime radical | C24 | C22×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
425 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)(7,10)(8,9)(11,12)(13,14,18,15,17,20,25,22)(16,19,24,21,23,26,28,27);(6,9)(7,11)(13,15,18,20,17,22,25,14)(16,21,24,26,23,27,28,19);(5,7,8,11)(6,10,9,12)(13,16,17,23)(14,19,20,26)(15,21,22,27)(18,24,25,28);(6,9)(10,12)(16,23)(19,26)(21,27)(24,28)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=e,f^2=h,g^2=h,b^a=be,d^a=deh,c^b=ceh,d^c=dh76 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#20554, #6370214, #6370215, … |
| maximal normal subgroup of | C2×#20554, #6370214, #6370215, … |
| maximal quotient of | #184482, #184483, #184494, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13542);