Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9103 |
| 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 | #386 |
| 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 | |
639 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 639 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:
(1,2)(3,4)(5,6,10,14,11,15,9,24)(7,16,18,12,19,28,25,27)(8,20,22,31,23,13,21,30)(17,33,34,26,35,32,36,29);(5,7)(6,12)(8,17)(9,25)(10,18)(11,19)(13,29)(14,28)(15,27)(16,24)(20,26)(21,36)(22,34)(23,35)(30,33)(31,32);(2,4)(5,8,11,23)(6,13,15,20)(7,17,19,35)(9,22,10,21)(12,26,27,29)(14,31,24,30)(16,32,28,33)(18,36,25,34)Polycyclic presentation:
2^8|a^2=g,c^2=h,d^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,c^b=cf,d^b=dh,d^c=dh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #94419, #94421, #133368, … |
| maximal normal subgroup of | #94419, #94421, #133368, … |
| maximal quotient of | #111227, #111228, #111554, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3978);