#62610

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

915 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)(19,25)(20,27)(22,28)(24,30)(26,31)(29,32);(1,3)(2,7)(4,11)(5,13)(6,12)(8,15)(9,10)(14,16)(17,18,21,23)(19,22,26,29)(20,24,27,30)(25,28,31,32);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,10,16)(17,19,20,25,21,26,27,31)(18,22,24,28,23,29,30,32);(3,12)(7,10)(11,13)(15,16)(19,26)(22,29)(25,31)(28,32);(1,5,6,4)(2,8,9,14)(3,13,12,11)(7,15,10,16)(18,23)(19,26)(24,30)(25,31)

Polycyclic presentation:

2^8|b^2=h,c^2=fg,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	#6367552, #6367555, #6367561, …
maximal normal subgroup of	#6367552, #6367555, #6367561, …
maximal quotient of	#6386969, #6389675, #6389692, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 55598);