#62546

order	256
nilpotency class	3
isoclinism	isoclinic class of #62253
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 ◁ C₄×#41 ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

920 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#6350565, #6350569, #6350592, …
maximal normal subgroup of	#6350565, #6350569, #6350592, …
maximal quotient of	#6386946, #6389635, #6389639, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 55534);