Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8325 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #379 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
635 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 635 conjugacy classes of subgroups — too many to draw. Only its 127 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)(7,16)(8,21,24,30)(9,25)(10,14)(11,15)(12,18)(13,23,28,22)(17,26,34,36)(19,27)(20,29)(31,33,35,32);(1,2)(3,4)(5,7,11,20)(6,12,15,27)(8,17,24,34)(9,19,10,18)(13,26,28,36)(14,29,25,16)(21,31,30,35)(22,33,23,32);(2,4)(5,8)(6,13)(7,17)(9,23)(10,22)(11,24)(12,26)(14,30)(15,28)(16,31)(18,33)(19,32)(20,34)(21,25)(27,36)(29,35)Polycyclic presentation:
2^8|a^2=g,b^2=h,d^2=h,e^2=h,b^a=bd,c^a=ce,d^a=dh,c^b=cf,d^b=dh,e^b=eh,d^c=dh,e^c=eh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #94443, #94444, #131981, … |
| maximal normal subgroup of | #94443, #94444, #131981, … |
| maximal quotient of | #111213, #111214, #111397, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3995);