Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3019 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×2−1+4 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1805 conjugacy classes of subgroups; only the 303 normal subgroups are shown.
This group has 1805 conjugacy classes of subgroups — too many to draw. Only its 303 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)(7,15)(8,12)(9,21)(10,13)(11,17)(14,24)(16,32)(18,29)(19,30)(20,27)(22,26)(23,28)(25,33)(31,34)(35,36);(5,7)(6,11)(8,16)(9,22)(10,18)(12,25)(13,26)(14,29)(15,21)(17,24)(19,35)(20,34)(23,33)(27,36)(28,31)(30,32);(5,8,10,20)(6,12,13,27)(7,16,18,34)(9,23,24,19)(11,25,26,36)(14,30,21,28)(15,31,29,32)(17,35,22,33);(6,13)(11,26)(12,27)(14,21)(15,29)(25,36)(28,30)(31,32)Polycyclic presentation:
2^8|a^2=g,c^2=h,e^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,e^b=eh,f^b=fh,e^c=eh58 conjugacy classes — view the full character table.
| maximal subgroup of | #147775, #359537, #359559, … |
| maximal normal subgroup of | #147775, #359537, #359559, … |
| maximal quotient of | #184931, #184937, #184943, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13605);