Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | Aut(G) order 65536 |
| 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 | |
898 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,3)(6,7)(10,14)(15,16)(17,18,21,20)(19,22,24,23);(1,2,5,3)(4,7,8,6)(17,19)(18,22)(20,23)(21,24);(1,2,5,3)(4,6,8,7)(9,10)(11,15)(12,16)(13,14)(17,18,21,20)(19,23,24,22);(1,3,5,2)(4,7,8,6)(9,11,13,12)(10,15,14,16);(1,4)(2,6)(3,7)(5,8)(9,12,13,11)(10,16,14,15)(17,20,21,18)(19,22,24,23)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=fh,d^2=g,e^2=fgh,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh58 conjugacy classes — view the full character table.
| maximal subgroup of | #97940, #166845, #166846, … |
| maximal normal subgroup of | #97940, #166845, #166846, … |
| maximal quotient of | #521458, #522220, #522618, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 32570);