#50524

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

736 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)(3,4)(5,7)(6,8)(9,10,13,16)(11,14,12,15)(17,18,22,25)(19,20,27,23)(21,29,30,24)(26,31,32,28);(3,5)(4,7)(17,19,22,27)(18,23,25,20)(21,26,30,32)(24,31,29,28);(1,3,6,5)(2,7,8,4)(9,10)(11,14)(12,15)(13,16)(17,20)(18,19)(21,28)(22,23)(24,26)(25,27)(29,32)(30,31);(1,4,6,7)(2,3,8,5)(9,11,13,12)(10,14,16,15)(17,19)(18,20)(21,26)(22,27)(23,25)(24,28)(29,31)(30,32);(1,5,6,3)(2,7,8,4)(9,12,13,11)(10,15,16,14)(17,21,22,30)(18,24,25,29)(19,26,27,32)(20,28,23,31)

Polycyclic presentation:

2^8|a^2=gh,b^2=f,c^2=h,d^2=fg,e^2=g,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×#50524, #7705857, #7706231, …
maximal normal subgroup of	C₂×#50524, #7705857, #7706231, …
maximal quotient of	#560785, #560922, #561886, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 43512);