Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2002 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #120 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
679 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4)(5,6,7,8)(9,10)(11,18)(12,23)(13,16)(14,17)(15,20)(19,22)(21,24);(9,11,14,22)(10,15,17,24)(12,21,13,20)(16,19,23,18);(2,4)(9,10)(11,19)(12,16)(13,23)(14,17)(15,20)(18,22)(21,24)Polycyclic presentation:
2^8|a^2=g,b^2=h,d^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,c^b=cf,d^b=dh,f^b=fh,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11017, #357570, #357572, … |
| maximal normal subgroup of | C2×#11017, #357570, #357572, … |
| maximal quotient of | #111453, #111454, #111500, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4005);