#62467

order	256
nilpotency class	3
isoclinism	isoclinic class of #62307
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #529 ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

943 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^8|c^2=g,d^2=h,e^2=f,g^2=h,b^a=bf,c^a=cg,e^a=ef,g^a=gh,d^b=df,e^b=eh,d^c=dh,e^c=eh

Conjugacy classes & character table

49 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#62467, #10579228, #10579237, …
maximal normal subgroup of	C₂×#62467, #10579228, #10579237, …
maximal quotient of	#6386905, #6389537, #6389556, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 55455);