Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D4×D16 |
| 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 | |
343 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 343 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)(5,6,12,8)(7,17,19,32)(9,23,26,15)(10,22,29,14)(11,24,28,16)(13,25,18,20)(21,35,30,33)(27,34,36,31);(5,7,12,19)(6,13,8,18)(9,25,26,20)(10,27,29,36)(11,30,28,21)(14,32,22,17)(15,33,23,35)(16,34,24,31);(2,4)(5,8,12,6)(7,18,19,13)(9,22,26,14)(10,23,29,15)(11,24,28,16)(17,20,32,25)(21,34,30,31)(27,35,36,33);(5,9)(6,14)(7,19)(8,22)(10,28)(11,29)(12,26)(13,18)(15,24)(16,23)(17,35)(20,36)(25,27)(32,33)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=dgh,e^a=eg,g^a=gh,d^b=de,e^b=eg,g^b=gh,e^d=eg,g^d=gh55 conjugacy classes — view the full character table.
| maximal subgroup of | #491713, #491717, #491723, … |
| maximal normal subgroup of | #491713, #491717, #491723, … |
| maximal quotient of | #491717, #491727, #491781, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26838);