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 | |
359 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 359 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) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,10,14)(7,15,19,33)(8,20,23,36)(9,24,22,12)(11,26,28,17)(13,30,32,21)(16,29,34,27)(18,35,25,31);(5,7)(6,11)(8,16)(9,25)(10,19)(12,29)(13,31)(14,28)(15,20)(17,21)(18,22)(23,34)(24,27)(26,30)(32,35)(33,36);(2,4);(6,12)(7,16)(8,21)(9,22)(11,15)(13,32)(14,24)(17,18)(19,34)(23,30)(25,26)(27,35)(28,33)(29,31)Polycyclic presentation:
2^8|a^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 | #513165, #513167, #513173, … |
| maximal normal subgroup of | #513165, #513167, #513173, … |
| maximal quotient of | #491716, #491721, #491780, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26834);