#39082

order	256
nilpotency class	2
isoclinism	isoclinic class of #37644
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	C₂⁴	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

728 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:

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

Polycyclic presentation:

2^8|a^2=g,b^2=f,c^2=fg,d^2=f,e^2=gh,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#6543767, #6543771, #6543777, …
maximal normal subgroup of	#6543767, #6543771, #6543777, …
maximal quotient of	#521546, #522243, #522780, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 32070);