Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
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×#121 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
437 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,4,3)(6,9)(10,12)(13,14,17,22)(15,23,25,19)(16,21,18,20)(24,28,26,27);(1,3,4,2)(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,26)(17,25)(18,24)(20,27)(21,28)(22,23);(5,7,8,11)(6,10,9,12)(14,20)(16,18)(19,27)(21,22)(23,28)(24,26);(6,9)(10,12)(13,16,17,18)(14,21,22,20)(15,24,25,26)(19,27,23,28)Polycyclic presentation:
2^8|a^2=g,b^2=gh,c^2=eh,d^2=h,f^2=h,b^a=be,c^a=cf,f^a=fh,d^b=de,d^c=dh,f^c=fh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#22931, #6381588, #6381599, … |
| maximal normal subgroup of | C2×#22931, #6381588, #6381599, … |
| maximal quotient of | #228285, #228301, #228349, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15919);