Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24468 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
726 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 726 conjugacy classes of subgroups — too many to draw. Only its 143 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:
(2,3)(5,6,11,15)(7,12,18,29)(8,19,21,32)(9,23,10,14)(13,25,31,22)(16,30,33,36)(17,28,24,27)(20,35,34,26);(1,2,4,3)(5,7)(6,12)(8,20)(9,17)(10,24)(11,18)(13,30)(14,27)(15,29)(16,22)(19,35)(21,34)(23,28)(25,33)(26,32)(31,36);(5,8,9,22,11,21,10,25)(6,13,14,19,15,31,23,32)(7,16,17,34,18,33,24,20)(12,26,27,30,29,35,28,36);(1,3,4,2)(8,21)(13,31)(16,33)(19,32)(20,34)(22,25)(26,35)(30,36)Polycyclic presentation:
2^8|a^2=h,b^2=e,c^2=f,d^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,g^b=gh,d^c=dh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #250391, #250393, #250399, … |
| maximal normal subgroup of | #250391, #250393, #250399, … |
| maximal quotient of | #297408, #297409, #297416, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18738);