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 | #3283 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
332 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 332 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)(5,6,11,8)(7,15,18,34)(9,22,26,33)(10,21,24,14)(12,29,19,17)(13,28,20,25)(16,35,27,31)(23,32,36,30);(5,7,11,18)(6,12,8,19)(9,23,26,36)(10,27,24,16)(13,31,20,35)(14,32,21,30)(15,33,34,22)(17,28,29,25);(2,4)(5,8,11,6)(7,12,18,19)(9,20,26,13)(10,21,24,14)(15,29,34,17)(16,30,27,32)(22,25,33,28)(23,31,36,35);(5,9)(6,13)(7,16)(8,20)(10,28)(11,26)(12,30)(14,33)(15,34)(17,29)(18,27)(19,32)(21,22)(24,25)Polycyclic presentation:
2^8|a^2=h,b^2=h,c^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 | #491736, #491739, #491740, … |
| maximal normal subgroup of | #491736, #491739, #491740, … |
| maximal quotient of | #491771, #491772, #491835, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26830);