Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29602 |
| automorphism | Aut(G) order 65536 |
| 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 | |
461 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 461 conjugacy classes of subgroups — too many to draw. Only its 119 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:
(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,19)(16,20)(17,22)(18,21)(23,32)(24,34)(25,33)(26,35)(27,37)(28,36)(29,39)(30,40)(31,38)(41,43)(42,44);(6,9)(10,12)(13,15,18,25)(14,19,21,33)(16,26,28,42)(17,29,31,24)(20,35,36,44)(22,39,38,34)(23,30,41,27)(32,40,43,37);(2,4)(5,7)(6,10)(8,11)(9,12)(13,16,18,28)(14,20,21,36)(15,23,25,41)(17,30,31,27)(19,32,33,43)(22,40,38,37)(24,42,29,26)(34,44,39,35);(7,11)(10,12)(14,21)(19,33)(20,36)(22,38)(32,43)(34,39)(35,44)(37,40)Polycyclic presentation:
2^8|a^2=f,b^2=h,c^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,c^b=cg,g^b=gh,d^c=de,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #142227, #142231, #142323, … |
| maximal normal subgroup of | #142227, #142231, #142323, … |
| maximal quotient of | #329599, #329600, #329601, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22712);