Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #121 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C2×#121 |
| prime radical | C24 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
383 conjugacy classes of subgroups; only the 167 normal subgroups are shown.
This group has 383 conjugacy classes of subgroups — too many to draw. Only its 167 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)(3,8)(4,6)(5,7)(9,10,13,18,15,20,14,19)(11,16,22,31,23,32,29,36)(12,24,27,35,28,34,26,17)(21,33,25,39,38,40,37,30);(1,3,5,4)(2,6,7,8)(9,11)(10,16)(12,25)(13,22)(14,29)(15,23)(17,33)(18,31)(19,36)(20,32)(21,26)(24,39)(27,38)(28,37)(30,34)(35,40);(1,4,5,3)(2,6,7,8)(9,12)(10,17)(11,21)(13,26)(14,27)(15,28)(16,30)(18,34)(19,24)(20,35)(22,37)(23,38)(25,29)(31,40)(32,39)(33,36);(9,13,15,14)(10,18,20,19)(11,22,23,29)(12,26,28,27)(16,31,32,36)(17,34,35,24)(21,37,38,25)(30,40,39,33)Polycyclic presentation:
2^8|a^2=f,b^2=e,c^2=e,d^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,c^b=cg,g^b=gh,f^c=fh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #137241, #137247, #142376, … |
| maximal normal subgroup of | #137241, #137247, #142376, … |
| maximal quotient of | #243151, #243156, #243160, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16662);