#41727

order	256
nilpotency class	2
isoclinism	isoclinic class of #40484
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

743 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)(3,5)(4,6)(7,8)(9,10,13,12)(11,15,16,14)(18,19)(20,27)(24,26)(28,32);(1,2)(3,7)(4,6)(5,8)(17,18,22,19)(20,23,27,25)(21,26,29,24)(28,31,32,30);(1,2,4,6)(3,5,8,7)(9,11,13,16)(10,14,12,15)(17,19,22,18)(20,23,27,25)(21,26,29,24)(28,30,32,31);(1,2)(3,5)(4,6)(7,8)(17,20,22,27)(18,23,19,25)(21,28,29,32)(24,30,26,31);(1,3,4,8)(2,5,6,7)(9,12,13,10)(11,15,16,14)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)

Polycyclic presentation:

2^8|a^2=fgh,b^2=f,c^2=g,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#41727, #7640180, #7640238, …
maximal normal subgroup of	C₂×#41727, #7640180, #7640238, …
maximal quotient of	#544841, #545277, #545799, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 34715);