Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35680 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| 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 | |
822 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
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)(19,26)(23,25)(27,32)(30,31);(1,3)(2,7)(4,6)(5,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,23)(20,25)(21,27)(22,26)(24,30)(28,31)(29,32);(9,11,13,12)(10,15,14,16)(17,20,22,18)(19,25,26,23)(21,24,29,28)(27,30,32,31);(1,4,5,7)(2,3,6,8)(9,12,13,11)(10,16,14,15)(17,21,22,29)(18,24,20,28)(19,27,26,32)(23,30,25,31)Polycyclic presentation:
2^8|a^2=h,b^2=f,d^2=fg,e^2=g,b^a=bf,c^a=cg,c^b=ch,e^d=eh64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#36016, #7629527, #7629533, … |
| maximal normal subgroup of | C2×#36016, #7629527, #7629533, … |
| maximal quotient of | #517636, #517637, #517638, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29004);