#47331

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

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:

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

Polycyclic presentation:

2^8|a^2=h,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×#47331, #7701271, #7701448, …
maximal normal subgroup of	C₂×#47331, #7701271, #7701448, …
maximal quotient of	#576968, #577002, #577101, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 40319);