Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | Aut(G) order 393216 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, prime radical, cosocle | C24 | C24 |
| commutator | C23 | C23×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
645 conjugacy classes of subgroups; only the 231 normal subgroups are shown.
This group has 645 conjugacy classes of subgroups — too many to draw. Only its 231 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)(6,9)(7,11)(13,14,17,15)(16,19,20,18)(22,23)(26,27);(14,15)(16,20)(21,22,25,23)(24,27,28,26);(1,3,4,2)(5,6)(7,10)(8,9)(11,12)(13,15,17,14)(16,19,20,18)(21,23,25,22)(24,27,28,26);(5,7,8,11)(6,10,9,12)(13,16,17,20)(14,18,15,19)(21,24)(22,26)(23,27)(25,28)Polycyclic presentation:
2^8|a^2=h,b^2=eg,c^2=egh,d^2=ef,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de64 conjugacy classes — view the full character table.
| Fitting subgroup of | #14601⋊C3 |
| maximal subgroup of | #136154, #136230, #137327, … |
| maximal normal subgroup of | #136154, #136230, #137327, … |
| maximal quotient of | #153981, #154011, #154048, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 7589);