Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1795 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #379 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
589 conjugacy classes of subgroups; only the 175 normal subgroups are shown.
This group has 589 conjugacy classes of subgroups — too many to draw. Only its 175 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 — 3 generators on 36 points:
(5,6,10,16,11,14,9,15)(7,17,19,30,20,12,25,27)(8,21,23,13,24,29,22,28)(18,32,33,26,34,36,35,31);(1,2,3,4)(5,7)(6,12)(8,18)(9,19)(10,25)(11,20)(13,26)(14,17)(15,27)(16,30)(21,36)(22,33)(23,35)(24,34)(28,31)(29,32);(5,8,11,24)(6,13,14,28)(7,18,20,34)(9,23,10,22)(12,26,17,31)(15,29,16,21)(19,35,25,33)(27,32,30,36)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=h,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,f^b=fh,e^c=eh,f^c=fh58 conjugacy classes — view the full character table.
| maximal subgroup of | #129786, #129788, #129830, … |
| maximal normal subgroup of | #129786, #129788, #129830, … |
| maximal quotient of | #105641, #105643, #105649, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3765);