#50931

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:

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

Polycyclic presentation:

2^8|a^2=gh,b^2=fg,c^2=fg,d^2=h,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×#50931, #7706691, #7706693, …
maximal normal subgroup of	C₂×#50931, #7706691, #7706693, …
maximal quotient of	#560360, #560554, #561543, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 43919);