#47347

order	256
nilpotency class	2
isoclinism	isoclinic class of #46350
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

688 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:

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

Polycyclic presentation:

2^8|a^2=gh,b^2=h,c^2=f,d^2=f,e^2=fg,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=df,e^b=eh,d^c=dh

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#47347, #7701290, #7701448, …
maximal normal subgroup of	C₂×#47347, #7701290, #7701448, …
maximal quotient of	#576872, #576874, #577133, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 40335);