#50628

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

734 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^8|a^2=gh,b^2=fg,c^2=gh,d^2=g,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×#50628, #7706012, #7706237, …
maximal normal subgroup of	C₂×#50628, #7706012, #7706237, …
maximal quotient of	#560804, #560916, #561904, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 43616);