Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | Aut(G) order 196608 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
942 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(3,7)(6,8)(9,10)(11,15)(12,13)(14,16)(19,23)(21,24);(2,4)(3,7)(10,12)(14,15)(17,18,20,22)(19,21,23,24);(1,2,5,4)(3,6,7,8)(9,10,13,12)(11,14,16,15)(17,19,20,23)(18,21,22,24);(1,3,5,7)(2,6,4,8)(9,11)(10,14)(12,15)(13,16);(1,4,5,2)(3,8,7,6)(9,12,13,10)(11,15,16,14)Polycyclic presentation:
2^8|b^2=fg,c^2=fgh,d^2=fh,e^2=h,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=dfh,e^b=eh52 conjugacy classes — view the full character table.
| Fitting subgroup of | #46333⋊C3 |
| maximal subgroup of | #7232011, #7232014, #7232037, … |
| maximal normal subgroup of | #7232011, #7232014, #7232037, … |
| maximal quotient of | #557086, #557118, #557370, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 39321);