#11999

order	256
nilpotency class	4
isoclinism	isoclinic class of #365
automorphism	Aut(G) order 8192
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#106
commutator	C₈	C₂²×C₈
Frattini, cosocle	C₄×C₈	C₂³
socle	C₂²	C₂×#106
prime radical	C₂³	#106
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

117 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^8|a^2=e,b^2=d,c^2=h,d^2=f,e^2=g,f^2=h,b^a=bd,d^a=df,f^a=fh,e^b=eh

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#94031, #94056, #129088, …
maximal normal subgroup of	#94031, #94056, #129088, …
maximal quotient of	#122919, #122922, #122925, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 4987);