Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24289 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#578 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
701 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 701 conjugacy classes of subgroups — too many to draw. Only its 143 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:
(2,4)(5,6,11,16)(7,13,19,27)(8,20,23,34)(9,24,10,15)(12,25,28,18)(14,22,30,21)(17,31,33,26)(29,35,36,32);(5,7,10,25,11,19,9,18)(6,12,24,27,16,28,15,13)(8,17,22,35,23,33,21,32)(14,26,34,36,30,31,20,29);(1,2,3,4)(5,7)(6,13)(8,17)(9,18)(10,25)(11,19)(12,24)(14,29)(15,28)(16,27)(20,31)(21,32)(22,35)(23,33)(26,34)(30,36);(5,8,10,22,11,23,9,21)(6,14,24,34,16,30,15,20)(7,17,25,35,19,33,18,32)(12,26,27,36,28,31,13,29)Polycyclic presentation:
2^8|a^2=h,b^2=eh,c^2=f,d^2=eh,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,g^b=gh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #501029, #501033, #501035, … |
| maximal normal subgroup of | #501029, #501033, #501035, … |
| maximal quotient of | #299255, #299260, #299263, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18794);