Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D4 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C23 | C4×D4 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
465 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,4)(5,6)(7,8)(9,10,15,17,14,18,13,19)(11,20,22,23,16,24,21,12);(1,2,3,4)(9,11,13,21,14,16,15,22)(10,12,17,20,18,23,19,24);(6,8)(9,12)(10,16)(11,18)(13,24)(14,23)(15,20)(17,21)(19,22)Polycyclic presentation:
2^8|a^2=g,b^2=d,d^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,d^c=dh,f^c=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11125, #357805, #357806, … |
| maximal normal subgroup of | C2×#11125, #357805, #357806, … |
| maximal quotient of | #116981, #116982, #116983, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4113);