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 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
358 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,8)(3,4,7,6)(9,10,12,14)(11,13,15,16)(17,18)(19,27)(20,25)(21,23)(22,26)(24,29)(28,31)(30,32);(1,3,5,7)(2,6,8,4)(9,11,12,15)(10,13,14,16)(18,23)(19,28)(20,22)(24,31)(27,32)(29,30);(1,4,5,6)(2,7,8,3)(10,14)(13,16)(17,19)(18,24)(20,28)(21,29)(22,30)(23,27)(25,32)(26,31);(2,8)(3,7)(10,14)(13,16)(17,20,21,22)(18,25,23,26)(19,28,29,30)(24,32,27,31)Polycyclic presentation:
2^8|a^2=ef,b^2=ef,c^2=eh,d^2=h,g^2=h,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,g^b=gh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #6634615, #6634618, #6634649, … |
| maximal normal subgroup of | #6634615, #6634618, #6634649, … |
| maximal quotient of | #383677, #383965, #384277, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 24761);