#46207

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	C₂⁴	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

676 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:

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

Polycyclic presentation:

2^8|a^2=h,b^2=fg,c^2=fg,d^2=fh,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×#46207, #7702777, #7702849, …
maximal normal subgroup of	C₂×#46207, #7702777, #7702849, …
maximal quotient of	#557211, #557223, #557539, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 39195);