#46222

order	256
nilpotency class	2
isoclinism	isoclinic class of #45438
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

690 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#46222, #7702777, #7702852, …
maximal normal subgroup of	C₂×#46222, #7702777, #7702852, …
maximal quotient of	#557180, #557220, #557476, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 39210);