Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17405 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×C22⋊C4 |
| commutator | C23 | C23×C4 |
| Frattini, cosocle | C24 | C24 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
647 conjugacy classes of subgroups; only the 183 normal subgroups are shown.
This group has 647 conjugacy classes of subgroups — too many to draw. Only its 183 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)(12,22,24,16)(13,17)(14,18)(19,33,35,27)(20,28)(21,29)(23,36,38,30)(25,31)(26,32)(34,40,37,39);(1,3)(2,5)(4,8)(6,7)(9,11,14,21)(10,15,18,29)(12,19,24,35)(13,25,26,20)(16,27,22,33)(17,31,32,28)(23,37,38,34)(30,40,36,39);(3,8)(5,7)(9,12)(10,16)(11,19)(13,23)(14,24)(15,27)(17,30)(18,22)(20,34)(21,35)(25,37)(26,38)(28,39)(29,33)(31,40)(32,36);(2,6)(5,7)(9,13,14,26)(10,17,18,32)(11,20,21,25)(12,23,24,38)(15,28,29,31)(16,30,22,36)(19,34,35,37)(27,39,33,40)Polycyclic presentation:
2^8|a^2=g,b^2=h,d^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce,d^b=dh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #135528, #139491, #139513, … |
| maximal normal subgroup of | #135528, #139491, #139513, … |
| maximal quotient of | #213513, #213516, #213518, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14739);