Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| 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 | |
769 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(3,7)(6,8)(9,10,12,13)(11,15,16,14)(17,18)(19,25)(20,28)(21,22)(23,27)(24,30)(26,31)(29,32);(1,2,5,4)(3,6,7,8)(10,13)(14,15)(18,22)(23,25)(24,28)(31,32);(1,3,5,7)(2,6,4,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,29)(21,27)(22,25)(24,32)(26,30)(28,31);(9,11)(10,14)(12,16)(13,15)(17,20)(18,24)(19,26)(21,30)(22,28)(23,31)(25,32)(27,29);(1,4,5,2)(3,8,7,6)(17,18)(19,23)(20,24)(21,22)(25,27)(26,31)(28,30)(29,32)Polycyclic presentation:
2^8|a^2=h,b^2=fg,c^2=fg,e^2=fg,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#41743, #7640031, #7640048, … |
| maximal normal subgroup of | C2×#41743, #7640031, #7640048, … |
| maximal quotient of | #544807, #545256, #545769, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 34731);