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 |
| prime radical | #3287 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
337 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 337 conjugacy classes of subgroups — too many to draw. Only its 109 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)(5,6,11,15)(7,16,17,35)(8,21,13,33)(9,24,27,23)(10,28,26,14)(12,30,31,19)(18,36,32,25)(20,22,29,34);(5,7,11,17)(6,12,15,31)(8,22,13,34)(9,25,27,36)(10,29,26,20)(14,18,28,32)(16,23,35,24)(19,33,30,21);(2,4)(7,17)(12,31)(16,35)(18,32)(19,30)(20,29)(22,34)(25,36);(5,8,11,13)(6,9,15,27)(7,18,17,32)(10,24,26,23)(12,20,31,29)(14,33,28,21)(16,30,35,19)(22,36,34,25)Polycyclic presentation:
2^8|a^2=h,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 | #491731, #491735, #491738, … |
| maximal normal subgroup of | #491731, #491735, #491738, … |
| maximal quotient of | #491763, #491774, #491827, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26832);