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, prime radical, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
746 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,6)(3,8)(10,14)(15,16)(17,18,22,20)(19,25,26,23)(21,24,29,28)(27,31,32,30);(1,2,5,6)(3,7,8,4)(9,10,13,14)(11,15,12,16)(19,26)(21,29)(23,25)(24,28);(1,3)(2,7)(4,6)(5,8)(9,11,13,12)(10,15,14,16)(17,19,22,26)(18,23,20,25)(21,27,29,32)(24,30,28,31);(9,11,13,12)(10,15,14,16)(17,20,22,18)(19,23,26,25)(21,28,29,24)(27,30,32,31);(1,4,5,7)(2,3,6,8)(9,12,13,11)(10,16,14,15)(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=f,c^2=fg,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 | #35498⋊C3 |
| maximal subgroup of | C2×#35498, #7628872, #7629103, … |
| maximal normal subgroup of | C2×#35498, #7628872, #7629103, … |
| maximal quotient of | #515460, #515632, #515745, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28486);