Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33758 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
296 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 296 conjugacy classes of subgroups — too many to draw. Only its 97 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)(11,20)(12,17)(13,25)(14,29)(15,19)(16,23)(18,27)(21,36)(22,37)(24,35)(26,32)(28,39)(30,34)(31,38)(33,40);(1,3)(2,5)(4,8)(6,7)(11,21)(16,31)(20,36)(22,30)(23,38)(26,40)(32,33)(34,37);(3,8)(5,7)(9,11)(10,16)(12,22)(13,26)(14,30)(15,21)(17,32)(18,34)(19,31)(20,25)(23,27)(24,38)(28,40)(29,33)(35,37)(36,39);(2,6)(5,7)(9,12,15,14)(10,17,19,29)(11,22,21,30)(13,24,28,27)(16,32,31,33)(18,25,35,39)(20,37,36,34)(23,26,38,40)Polycyclic presentation:
2^8|d^2=h,f^2=gh,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fg,g^a=gh,c^b=ceh,f^c=fg,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #151184, #151492, C2×#33763, … |
| maximal normal subgroup of | #151184, #151492, C2×#33763, … |
| maximal quotient of | #496266, #496675, #496817, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26751);