#32857

order	256
nilpotency class	3
isoclinism	isoclinic class of #32712
automorphism	Aut(G) order 65536
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

255 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

(2,3)(6,7)(9,10,13,16)(11,17,19,14)(12,15,21,24)(18,23,20,22)(25,26,28,30)(27,33,35,31)(29,32,36,38)(34,39,37,40);(1,2,5,3)(4,6,8,7)(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,23,24,22)(26,30)(27,34)(29,36)(31,39)(33,40)(35,37);(1,3,5,2)(4,6,8,7)(12,21)(15,24)(18,20)(22,23)(25,27,28,35)(26,31,30,33)(29,37,36,34)(32,39,38,40);(1,4,5,8)(2,6,3,7)(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23)(26,30)(27,35)(32,38)(34,37)

Polycyclic presentation:

2^8|a^2=efh,b^2=eh,c^2=f,d^2=eh,g^2=h,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=def,g^b=gh,d^c=de,g^c=gh

Conjugacy classes & character table

34 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#32857, #6631590, #6631593, …
maximal normal subgroup of	C₂×#32857, #6631590, #6631593, …
maximal quotient of	#391399, #391436, #391495, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 25845);