#50942

order	256
nilpotency class	2
isoclinism	isoclinic class of #47826
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

695 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:

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

Polycyclic presentation:

2^8|a^2=fh,b^2=f,c^2=f,d^2=h,e^2=g,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfh,e^b=eh,d^c=dh

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#50942, #7706655, #7706670, …
maximal normal subgroup of	C₂×#50942, #7706655, #7706670, …
maximal quotient of	#560860, #560956, #561943, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 43930);