#62218

order	256
nilpotency class	3
isoclinism	isoclinic class of #61629
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

849 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^8|a^2=fh,b^2=h,d^2=h,e^2=fh,g^2=h,b^a=bf,c^a=cg,e^a=ef,g^a=gh,d^b=dfh,e^b=eh,g^c=gh

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#6364555, #6364559, #6364587, …
maximal normal subgroup of	#6364555, #6364559, #6364587, …
maximal quotient of	#6386459, #6388571, #6388595, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 55206);