Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #58990 |
| automorphism | Aut(G) order 98304 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | #392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
825 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,6)(4,8)(10,13)(14,16);(1,2,5,6)(3,4,7,8)(10,13)(14,16)(19,23)(21,24);(9,10)(11,14)(12,13)(15,16)(17,18,20,22)(19,21,23,24);(1,3,5,7)(2,4,6,8)(9,10)(11,14)(12,13)(15,16)(17,19,20,23)(18,21,22,24);(1,4)(2,7)(3,6)(5,8)(9,11,12,15)(10,14,13,16)Polycyclic presentation:
2^8|b^2=f,c^2=gh,d^2=fgh,e^2=g,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh64 conjugacy classes — view the full character table.
| Fitting subgroup of | #60026⋊C3 |
| maximal subgroup of | C2×#60026, #7666916, #7666939, … |
| maximal normal subgroup of | C2×#60026, #7666916, #7666939, … |
| maximal quotient of | #532151, #532674, #532833, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53014);