Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30396 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #2011 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
475 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6)(7,9)(8,10)(11,12)(13,14)(15,22)(16,18)(17,19)(20,26)(21,25)(23,28)(24,27);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(14,18)(15,23)(19,21)(24,26);(2,4)(5,8,9,12)(6,10,7,11)(14,19)(17,25)(18,21)(20,28)(22,27)(24,26);(6,7)(10,11)(13,15,16,23)(14,20,18,27)(17,26,25,24)(19,28,21,22)Polycyclic presentation:
2^8|b^2=eh,c^2=eh,d^2=h,f^2=h,b^a=be,c^a=cf,d^a=def,f^a=fh,c^b=cg,d^b=de,f^c=fh,f^d=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #6633229, #6633234, #6633250, … |
| maximal normal subgroup of | #6633229, #6633234, #6633250, … |
| maximal quotient of | #337676, #337678, #337684, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23728);