#51273

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

701 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)(9,10,14,18)(11,12,16,15)(13,17,21,24)(19,20,23,22)(25,26)(27,31)(28,32)(29,30)(33,35)(34,39)(36,40)(37,38);(1,2,5,4)(3,6,7,8)(11,16)(12,15)(13,21)(17,24)(26,29)(27,33)(32,37)(34,36);(1,3)(2,6)(4,8)(5,7)(9,11)(10,15)(12,18)(13,19)(14,16)(17,22)(20,24)(21,23)(25,27,30,33)(26,31,29,35)(28,36,38,34)(32,40,37,39);(9,12,14,15)(10,16,18,11)(13,20,21,22)(17,23,24,19)(25,28,30,38)(26,32,29,37)(27,34,33,36)(31,39,35,40);(1,4,5,2)(3,8,7,6)(9,13,14,21)(10,17,18,24)(11,19,16,23)(12,20,15,22)(25,29,30,26)(27,35,33,31)(28,37,38,32)(34,40,36,39)

Polycyclic presentation:

2^8|a^2=fh,b^2=fgh,c^2=f,d^2=h,e^2=fg,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×#51273, #7707259, #7707281, …
maximal normal subgroup of	C₂×#51273, #7707259, #7707281, …
maximal quotient of	#560843, #561075, #561947, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 44261);