Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | Aut(G) order 131072 |
| 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 | #2856 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
992 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,5)(6,8)(9,10,14,11)(12,15,13,16)(17,18,21,19)(20,23,24,22);(2,5)(3,7)(9,11,14,10)(12,15,13,16)(18,19)(22,23);(1,2)(3,6)(4,5)(7,8)(9,12,14,13)(10,15,11,16)(17,19,21,18)(20,23,24,22);(17,20)(18,22)(19,23)(21,24);(1,3,4,7)(2,6,5,8)(9,13,14,12)(10,16,11,15)Polycyclic presentation:
2^8|a^2=gh,b^2=fgh,c^2=gh,e^2=fh,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 | #97833, #97949, #166648, … |
| maximal normal subgroup of | #97833, #97949, #166648, … |
| maximal quotient of | #521497, #522143, #522643, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 32756);