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, prime radical | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
321 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
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)(19,22)(20,24)(21,25)(23,26)(27,32)(28,30)(29,31);(1,3,4,8)(2,5,6,7)(10,13)(14,15)(17,19,26,27,21,29,20,28)(18,22,23,32,25,31,24,30);(10,13)(14,15)(17,19,21,29)(18,22,25,31)(20,27,26,28)(23,30,24,32);(2,6)(5,7)(9,11,12,16)(10,14,13,15)(17,20,21,26)(18,23,25,24)(19,27,29,28)(22,30,31,32)Polycyclic presentation:
2^8|a^2=e,b^2=efgh,c^2=h,d^2=fh,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×#29038, #6460498, #6460513, … |
| maximal normal subgroup of | C2×#29038, #6460498, #6460513, … |
| maximal quotient of | #362529, #363688, #367682, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22026);