Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35346 |
| 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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1033 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,6,4)(3,7,8,5)(9,10,13,15)(11,14,16,12)(17,18,21,20)(19,22,24,23);(1,3)(2,7)(4,5)(6,8)(9,11)(10,12)(13,16)(14,15)(18,20)(22,23);(2,4)(3,8)(9,11,13,16)(10,12,15,14)(17,18,21,20)(19,23,24,22);(1,4,6,2)(3,5,8,7)(9,11)(10,14)(12,15)(13,16)(17,19,21,24)(18,22,20,23);(1,5)(2,3)(4,8)(6,7)(9,12,13,14)(10,11,15,16)(17,20,21,18)(19,23,24,22)Polycyclic presentation:
2^8|a^2=h,c^2=f,d^2=fg,e^2=f,b^a=bf,c^a=cg,c^b=ch,e^b=eh,d^c=dh64 conjugacy classes — view the full character table.
| Fitting subgroup of | #35481⋊C3 |
| maximal subgroup of | C2×#35481, #7628855, #7629006, … |
| maximal normal subgroup of | C2×#35481, #7628855, #7629006, … |
| maximal quotient of | #515459, #515637, #515724, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28469);