#52660

order	256
nilpotency class	2
isoclinism	isoclinic class of #52308
automorphism	Aut(G) = C₂¹⁵ (order 32768) · Inn(G) = C₂⁵ · Out(G) = C₂¹⁰
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

689 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

(3,7)(6,8)(9,10,14,12)(11,17,18,15)(13,20,21,16)(19,22,24,23)(25,26,29,32)(27,33,35,30)(28,31,36,38)(34,39,40,37);(1,2)(3,6)(4,5)(7,8)(10,12)(11,18)(16,20)(19,24)(25,26,29,32)(27,30,35,33)(28,31,36,38)(34,37,40,39);(2,5)(6,8)(9,11,14,18)(10,15,12,17)(13,19,21,24)(16,22,20,23)(25,26)(27,30)(28,31)(29,32)(33,35)(34,37)(36,38)(39,40);(1,2)(3,6)(4,5)(7,8)(9,12,14,10)(11,15,18,17)(13,20,21,16)(19,22,24,23)(25,27)(26,30)(28,34)(29,35)(31,37)(32,33)(36,40)(38,39);(1,3)(2,6)(4,7)(5,8)(9,13,14,21)(10,16,12,20)(11,19,18,24)(15,22,17,23)(25,28,29,36)(26,31,32,38)(27,34,35,40)(30,37,33,39)

Polycyclic presentation:

2^8|a^2=g,b^2=fg,c^2=f,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=dfg,e^a=efgh,c^b=ch,d^b=df,d^c=dfgh

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#52660, #8092253, #8092474, …
maximal normal subgroup of	C₂×#52660, #8092253, #8092474, …
maximal quotient of	#582277, #582463, #582485, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 45648);