Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23387 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
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 | |
681 conjugacy classes of subgroups; only the 175 normal subgroups are shown.
This group has 681 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) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,4)(9,10,15,20)(11,21,25,32)(12,26,28,36)(13,19,14,18)(16,24,33,23)(17,30,35,29)(22,38,39,31)(27,34,40,37);(5,6)(7,8)(9,11,15,25)(10,16,20,33)(12,27,28,40)(13,24,14,23)(17,34,35,37)(18,32,19,21)(22,29,39,30)(26,38,36,31);(2,4)(6,8)(9,12,13,29,15,28,14,30)(10,17,18,26,20,35,19,36)(11,22,23,40,25,39,24,27)(16,31,21,37,33,38,32,34);(12,28)(17,35)(22,39)(26,36)(27,40)(29,30)(31,38)(34,37)Polycyclic presentation:
2^8|a^2=h,b^2=h,c^2=e,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,g^a=gh,c^b=cg,e^b=eh,f^b=fh,g^b=gh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #142285, #142287, #394639, … |
| maximal normal subgroup of | #142285, #142287, #394639, … |
| maximal quotient of | #238877, #238881, #238886, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16530);