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) = C25 |
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×#127 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
559 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)(6,10)(11,12)(13,14)(15,22)(16,20)(17,19)(18,23)(21,25)(24,26)(27,28);(5,6,9,10)(7,11,8,12)(13,15,16,23)(14,18,20,22)(17,24,25,28)(19,27,21,26);(5,7,9,8)(6,11,10,12)(14,19)(15,23)(17,25)(18,26)(20,21)(22,27);(5,8,9,7)(6,11,10,12)(15,23)(18,22)(24,28)(26,27)Polycyclic presentation:
2^8|a^2=g,b^2=e,c^2=eh,d^2=eh,f^2=h,b^a=be,c^a=cf,f^a=fh,c^b=ch,d^b=de,f^c=fh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#22549, #6379453, #6379457, … |
| maximal normal subgroup of | C2×#22549, #6379453, #6379457, … |
| maximal quotient of | #224548, #224560, #224572, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15537);