Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | Aut(G) = C23×#1202 (order 8192) · Inn(G) = C25 · Out(G) = C25×D4 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C4 | C25 |
| commutator, socle | C22 | C24×C4 |
| prime radical | #3480 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
953 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10,13,17,14,16,22,23)(11,15,19,12,18,20,24,21);(2,5)(6,8)(9,11)(10,15)(12,17)(13,19)(14,18)(16,20)(21,23)(22,24);(9,12,13,20,14,21,22,15)(10,11,17,19,16,18,23,24);(1,3,4,7)(2,6,5,8)(9,11)(10,15)(12,17)(13,19)(14,18)(16,20)(21,23)(22,24);(10,16)(11,18)(17,23)(19,24)Polycyclic presentation:
2^8|a^2=g,c^2=g,d^2=f,g^2=h,b^a=bf,c^a=ch,e^a=eh,e^b=eh,e^d=eh88 conjugacy classes — view the full character table.
| maximal subgroup of | #129548, #210278, #210279, … |
| maximal normal subgroup of | #129548, #210278, #210279, … |
| maximal quotient of | #6369998, #6369999, #6370043, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53207);