Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | Aut(G) order 524288 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C23 | C22×D4 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
475 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)(15,21)(16,20)(17,22)(18,23)(19,24)(25,28)(26,27);(6,9)(10,12)(13,15,18,25,19,26,17,16)(14,20,22,27,24,28,23,21);(5,7,8,11)(6,10,9,12)(13,16,17,26,19,25,18,15)(14,21,23,28,24,27,22,20);(13,17,19,18)(14,22,24,23)(15,16,26,25)(20,27,28,21)Polycyclic presentation:
2^8|a^2=g,b^2=f,c^2=e,d^2=h,e^2=h,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh88 conjugacy classes — view the full character table.
| maximal subgroup of | #109025, #109027, #109029, … |
| maximal normal subgroup of | #109025, #109027, #109029, … |
| maximal quotient of | #191164, #191172, #191180, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13748);