Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29736 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
451 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 451 conjugacy classes of subgroups — too many to draw. Only its 127 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 44 points:
(5,6)(7,10)(8,9)(11,12)(13,14,18,23,19,24,17,22)(15,20,26,34,27,35,31,37)(16,28,29,36,30,38,32,21)(25,39,41,44,42,43,40,33);(1,2,3,4)(6,9)(10,12)(13,15)(14,20)(16,25)(17,31)(18,26)(19,27)(21,33)(22,37)(23,34)(24,35)(28,39)(29,41)(30,42)(32,40)(36,44)(38,43);(2,4)(5,7)(6,10)(8,11)(9,12)(13,16,19,30)(14,21,24,36)(15,25,27,42)(17,29,18,32)(20,33,35,44)(22,28,23,38)(26,40,31,41)(34,43,37,39);(7,11)(10,12)(13,17,19,18)(14,22,24,23)(15,26,27,31)(16,29,30,32)(20,34,35,37)(21,28,36,38)(25,40,42,41)(33,43,44,39)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=h,d^2=h,f^2=h,b^a=be,c^a=cf,c^b=cg,d^b=dh,d^c=de,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #142800, #142803, #142809, … |
| maximal normal subgroup of | #142800, #142803, #142809, … |
| maximal quotient of | #331120, #331130, #331140, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23245);