#42785

order	256
nilpotency class	2
isoclinism	isoclinic class of #40484
automorphism	Aut(G) order 65536
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁵
prime radical	C₂²×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

742 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:

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

Polycyclic presentation:

2^8|a^2=fgh,b^2=g,c^2=fg,d^2=fg,e^2=fh,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#6696332, #6696337, #6696397, …
maximal normal subgroup of	#6696332, #6696337, #6696397, …
maximal quotient of	#544604, #545144, #545632, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 35773);