Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23903 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C24 | C24 |
| prime radical | #534 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
605 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 605 conjugacy classes of subgroups — too many to draw. Only its 123 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,15)(12,16)(13,17)(14,18)(19,29)(20,27)(21,28)(22,30)(23,31)(24,32)(25,33)(26,35)(34,39)(36,38)(37,40);(1,3,4,8)(2,5,6,7)(9,11)(10,15)(12,19)(13,20)(14,21)(16,29)(17,27)(18,28)(22,34)(23,35)(24,36)(25,37)(26,31)(30,39)(32,38)(33,40);(9,12,14,23)(10,16,18,31)(11,19,21,35)(13,24,25,22)(15,26,28,29)(17,32,33,30)(20,36,37,34)(27,39,40,38);(2,6)(5,7)(9,13,14,25)(10,17,18,33)(11,20,21,37)(12,22,23,24)(15,27,28,40)(16,30,31,32)(19,34,35,36)(26,38,29,39)Polycyclic presentation:
2^8|b^2=e,c^2=h,d^2=h,b^a=be,c^a=cf,d^a=de,g^a=gh,c^b=cg,f^b=fh,d^c=dh40 conjugacy classes — view the full character table.
| maximal subgroup of | #142545, #142547, #142577, … |
| maximal normal subgroup of | #142545, #142547, #142577, … |
| maximal quotient of | #251226, #251227, #251228, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16977);