Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23444 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C24 | C24 |
| prime radical | #3449 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
997 conjugacy classes of subgroups; only the 183 normal subgroups are shown.
This group has 997 conjugacy classes of subgroups — too many to draw. Only its 183 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)(7,11)(8,12)(9,13)(10,14)(15,22)(16,23)(17,24)(18,26)(19,27)(20,28)(21,29)(25,31)(30,34)(32,35)(33,36);(5,7,10,17)(6,11,14,24)(8,15,19,31)(9,20,21,16)(12,25,27,22)(13,28,29,23)(18,32,33,30)(26,34,36,35);(2,4)(5,8)(6,12)(7,15)(9,18)(10,19)(11,22)(13,26)(14,27)(16,30)(17,31)(20,32)(21,33)(23,34)(24,25)(28,35)(29,36);(5,9,10,21)(6,13,14,29)(7,16,17,20)(8,18,19,33)(11,23,24,28)(12,26,27,36)(15,30,31,32)(22,34,25,35)Polycyclic presentation:
2^8|b^2=h,d^2=h,b^a=be,c^a=cf,g^a=gh,c^b=cg,d^b=dh,e^c=eh46 conjugacy classes — view the full character table.
| maximal subgroup of | #142225, #142227, #142417, … |
| maximal normal subgroup of | #142225, #142227, #142417, … |
| maximal quotient of | #232326, #232327, #232328, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16434);