#62574

order	256
nilpotency class	3
isoclinism	isoclinic class of #62283
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

854 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)(3,10)(4,8)(5,14)(6,9)(7,12)(11,16)(13,15)(17,18)(19,23)(20,29)(21,25)(22,27)(24,31)(26,30)(28,32);(1,3)(2,7)(4,11)(5,13)(6,12)(8,15)(9,10)(14,16)(17,19)(18,23)(20,26)(21,28)(22,24)(25,32)(27,31)(29,30);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,10,16)(17,20,30,23,22,32,21,31)(18,24,25,28,27,19,26,29);(3,12)(7,10)(11,13)(15,16)(17,21,22,30)(18,25,27,26)(19,28,24,29)(20,23,32,31);(1,5,6,4)(2,8,9,14)(3,13,12,11)(7,15,10,16)(17,21,22,30)(18,25,27,26)(19,29,24,28)(20,31,32,23)

Polycyclic presentation:

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

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#250488, #250492, #250496, …
maximal normal subgroup of	#250488, #250492, #250496, …
maximal quotient of	#6386959, #6389654, #6389682, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 55562);