Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33830 |
| automorphism | Aut(G) order 8192 |
| 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 ◁ C4×D4 ◁ C8×D4 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
334 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 334 conjugacy classes of subgroups — too many to draw. Only its 107 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) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(6,11)(7,15)(8,19)(9,22)(12,27)(13,30)(14,33)(16,29)(17,31)(18,25)(20,24)(23,26)(28,32)(34,35);(5,6,10,12)(7,16,18,30)(8,20,23,33)(9,24,22,14)(11,26,27,19)(13,31,29,17)(15,35,25,28)(21,34,36,32);(2,4)(6,12)(11,27)(13,29)(14,24)(16,30)(20,33)(28,35)(32,34);(5,7,10,18)(6,13,12,29)(8,21,23,36)(9,25,22,15)(11,28,27,35)(14,34,24,32)(16,33,30,20)(17,26,31,19)Polycyclic presentation:
2^8|b^2=h,d^2=h,e^2=gh,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eg,g^a=gh,c^b=ch,d^b=de,e^b=eg,g^b=gh,e^d=eg,g^d=gh43 conjugacy classes — view the full character table.
| maximal subgroup of | #151521, #151523, #151575, … |
| maximal normal subgroup of | #151521, #151523, #151575, … |
| maximal quotient of | #491767, #491770, #491831, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26831);