Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3055 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
401 conjugacy classes of subgroups; only the 175 normal subgroups are shown.
This group has 401 conjugacy classes of subgroups — too many to draw. Only its 175 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,14,19,24)(15,21,25,28)(16,20,26,27)(17,23,18,22);(6,9)(10,12)(13,15,17,16,19,25,18,26)(14,20,22,21,24,27,23,28);(5,7,8,11)(6,10,9,12)(13,16,17,25,19,26,18,15)(14,21,22,27,24,28,23,20);(1,3,4,2)(15,25)(16,26)(20,27)(21,28)Polycyclic presentation:
2^8|a^2=g,b^2=fh,c^2=e,d^2=gh,e^2=h,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,c^b=ch,d^b=dh,d^c=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #100114, #100120, #124621, … |
| maximal normal subgroup of | #100114, #100120, #124621, … |
| maximal quotient of | #212617, #212623, #212629, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14645);