#50343

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

697 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:

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

Polycyclic presentation:

2^8|a^2=fgh,b^2=g,c^2=fgh,d^2=fg,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×#50343, #7706143, #7706221, …
maximal normal subgroup of	C₂×#50343, #7706143, #7706221, …
maximal quotient of	#561192, #561264, #562276, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 43331);