#51430

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

698 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:

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

Polycyclic presentation:

2^8|a^2=gh,b^2=g,c^2=h,d^2=fh,e^2=f,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×#51430, #7707049, #7707057, …
maximal normal subgroup of	C₂×#51430, #7707049, #7707057, …
maximal quotient of	#560458, #560615, #561467, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 44418);