#62618

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

883 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)(19,26)(20,21)(22,25)(23,30)(24,29)(27,28)(31,32);(1,3)(2,7)(4,11)(5,13)(6,12)(8,15)(9,10)(14,16)(17,19,22,29)(18,24,25,26)(20,27,30,32)(21,31,23,28);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,10,16)(17,20,21,18,22,30,23,25)(19,27,31,24,29,32,28,26);(3,12)(7,10)(11,13)(15,16)(17,21,22,23)(18,20,25,30)(19,28,29,31)(24,32,26,27);(1,5,6,4)(2,8,9,14)(3,13,12,11)(7,15,10,16)(18,25)(19,29)(20,30)(28,31)

Polycyclic presentation:

2^8|b^2=h,c^2=fg,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,e^c=eh

Conjugacy classes & character table

49 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#6367550, #6367557, #6367559, …
maximal normal subgroup of	#6367550, #6367557, #6367559, …
maximal quotient of	#6386969, #6389675, #6389688, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 55606);