Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3019 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
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 | C2×2−1+4 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
661 conjugacy classes of subgroups; only the 187 normal subgroups are shown.
This group has 661 conjugacy classes of subgroups — too many to draw. Only its 187 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:
(1,2,4,3)(5,6,10,14)(7,15,18,29)(8,12,19,27)(9,21,24,13)(11,22,26,17)(16,32,33,31)(20,30,23,28)(25,35,36,34);(5,7)(6,11)(8,16)(9,22)(10,18)(12,25)(13,29)(14,26)(15,21)(17,24)(19,33)(20,35)(23,34)(27,36)(28,31)(30,32);(5,8,10,19)(6,12,14,27)(7,16,18,33)(9,23,24,20)(11,25,26,36)(13,30,21,28)(15,31,29,32)(17,35,22,34);(1,3,4,2)(8,19)(12,27)(16,33)(20,23)(25,36)(28,30)(31,32)(34,35)Polycyclic presentation:
2^8|a^2=g,c^2=h,d^2=gh,e^2=h,b^a=be,c^a=cf,e^a=eh,e^b=eh,f^b=fh,d^c=dh,e^c=eh58 conjugacy classes — view the full character table.
| maximal subgroup of | #495760, #495761, #495764, … |
| maximal normal subgroup of | #495760, #495761, #495764, … |
| maximal quotient of | #208747, #208755, #208763, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14464);