Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3198 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#537 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
695 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,3,4)(5,6)(7,11)(8,10)(9,12)(13,14)(15,19)(16,21)(17,26)(18,22)(20,23)(24,28)(25,27);(5,7)(6,9)(8,12)(10,11)(13,15)(14,19)(16,25)(17,23)(18,24)(20,26)(21,27)(22,28);(7,12)(9,11)(13,15)(14,20)(16,23)(17,25)(18,24)(19,26)(21,28)(22,27);(6,10)(9,11)(13,16,18,17)(14,21,22,26)(15,23,24,25)(19,20,28,27)Polycyclic presentation:
2^8|a^2=g,d^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce,d^b=dh,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#21772, #6374163, #6374166, … |
| maximal normal subgroup of | C2×#21772, #6374163, #6374166, … |
| maximal quotient of | #213700, #213701, #213705, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14760);