Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33522 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
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 ◁ #117 ◁ #405 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
374 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 374 conjugacy classes of subgroups — too many to draw. Only its 119 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,11)(8,12)(9,22)(10,15)(13,19)(14,20)(16,24)(17,23)(18,32)(21,28)(25,31)(26,30)(27,33)(29,34)(35,36);(1,2,3,4)(5,7)(6,11)(8,17)(9,18)(10,16)(12,23)(13,25)(14,26)(15,24)(19,31)(20,30)(21,29)(22,32)(27,35)(28,34)(33,36);(6,12)(8,19)(9,20)(11,23)(13,14)(15,28)(17,31)(18,30)(21,33)(22,27)(24,34)(25,26)(29,36)(32,35);(7,16)(11,24)(17,29)(18,30)(23,34)(25,35)(26,32)(31,36)Polycyclic presentation:
2^8|b^2=e,f^2=gh,g^2=h,b^a=be,c^a=cf,f^a=fg,g^a=gh,d^b=dh,f^c=fg,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #500749, #500754, #501020, … |
| maximal normal subgroup of | #500749, #500754, #501020, … |
| maximal quotient of | #490434, #490440, #490458, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26573);