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 | #3475 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
957 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10,14,11,13,15,21,17)(12,16,20,18,19,22,24,23);(2,5)(6,8)(10,15)(11,17)(16,22)(18,23);(9,11,14,15,13,17,21,10)(12,18,20,22,19,23,24,16);(1,3,4,7)(2,6,5,8)(9,12,13,19)(10,16,15,22)(11,18,17,23)(14,20,21,24);(12,19)(16,22)(18,23)(20,24)Polycyclic presentation:
2^8|a^2=g,c^2=g,d^2=f,g^2=h,b^a=bf,c^a=ch,c^b=ch,e^d=eh88 conjugacy classes — view the full character table.
| maximal subgroup of | #129546, #210282, #210283, … |
| maximal normal subgroup of | #129546, #210282, #210283, … |
| maximal quotient of | #6369999, #6370004, #6370039, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53205);