Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | Aut(G) = C213 (order 8192) · Inn(G) = C24 · Out(G) = C29 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C24 |
| commutator, socle | C23 | C23×C4 |
| prime radical | C24 | C22×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
338 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18,21,20,22,24,30,29)(19,23,27,28,25,31,32,26);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(17,19)(18,23)(20,28)(21,27)(22,25)(24,31)(26,29)(30,32);(1,4,5,3)(2,6,7,8)(10,14)(13,16)(19,25)(23,31)(26,28)(27,32);(17,20,21,24,22,29,30,18)(19,26,27,23,25,28,32,31)Polycyclic presentation:
2^8|a^2=g,b^2=ef,c^2=e,d^2=g,g^2=h,b^a=be,c^a=cf,d^a=dh,c^b=ch,d^b=dh76 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#21671, #6373562, #6373564, … |
| maximal normal subgroup of | C2×#21671, #6373562, #6373564, … |
| maximal quotient of | #354392, #354393, #354394, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14659);