Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35346 |
| automorphism | Aut(G) order 786432 |
| 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 | C25 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
927 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 32 points:
(3,7)(5,8)(10,13)(11,15)(17,18,22,23)(19,25,26,20)(21,24,29,30)(27,31,32,28);(1,2,4,6)(3,5,7,8)(9,10)(11,14)(12,13)(15,16)(18,23)(19,26)(21,29)(28,31);(1,3)(2,5)(4,7)(6,8)(9,11,12,15)(10,14,13,16)(17,19,22,26)(18,20,23,25)(21,27,29,32)(24,28,30,31);(17,20)(18,19)(21,28)(22,25)(23,26)(24,27)(29,31)(30,32);(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,31)(26,32)Polycyclic presentation:
2^8|a^2=h,b^2=fg,c^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 | #35456⋊C3 |
| maximal subgroup of | #97560, #129483, #134214, … |
| maximal normal subgroup of | #97560, #129483, #134214, … |
| maximal quotient of | #515291, #515583, #515993, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28444);