Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| 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 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
330 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,6)(3,7,8,5)(9,10,12,13)(11,15,16,14)(17,18)(19,24)(20,28)(21,30)(22,26)(23,25)(27,32)(29,31);(1,3,4,8)(2,5,6,7)(10,13)(14,15)(17,19,21,29,22,20,23,27)(18,24,30,31,26,28,25,32);(10,13)(14,15)(17,20,22,19)(18,24,26,28)(21,29,23,27)(25,31,30,32);(2,6)(5,7)(9,11,12,16)(10,14,13,15)(17,21,22,23)(18,25,26,30)(19,27,20,29)(24,31,28,32)Polycyclic presentation:
2^8|a^2=e,b^2=efg,c^2=h,d^2=f,g^2=h,b^a=be,c^a=cf,d^a=deh,c^b=cg,d^b=dh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#29033, #6460498, #6460513, … |
| maximal normal subgroup of | C2×#29033, #6460498, #6460513, … |
| maximal quotient of | #362529, #363690, #367681, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22021);