#50520

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

694 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:

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

Polycyclic presentation:

2^8|a^2=h,b^2=f,c^2=fh,d^2=g,e^2=fg,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×#50520, #7706049, #7706230, …
maximal normal subgroup of	C₂×#50520, #7706049, #7706230, …
maximal quotient of	#560820, #561043, #561908, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 43508);