#8631

order	256
nilpotency class	3
isoclinism	isoclinic class of #8626
automorphism	Aut(G) order 16384
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#380
commutator	C₂²×C₄	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
prime radical	C₂³	#117
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

160 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2,4,8,5,7,13,15)(3,9,10,16,11,6,12,14)(17,18)(19,28)(20,33,35,40)(21,36)(22,26)(23,27)(24,30)(25,34,41,37)(29,38,45,48)(31,39)(32,43)(42,44,47,46);(17,19,21,30,23,32,22,31)(18,24,26,28,27,39,36,43)(20,29,34,44,35,45,37,46)(25,38,40,42,41,48,33,47);(1,3,5,11)(2,6,7,9)(4,12,13,10)(8,16,15,14)(17,20)(18,25)(19,29)(21,34)(22,37)(23,35)(24,38)(26,40)(27,41)(28,42)(30,44)(31,46)(32,45)(33,36)(39,48)(43,47)

Polycyclic presentation:

2^8|a^2=f,b^2=d,c^2=h,d^2=g,e^2=g,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,e^c=eg,f^c=fgh

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#94481, #130458, #131082, …
maximal normal subgroup of	#94481, #130458, #131082, …
maximal quotient of	#98837, #98845, #98851, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 1619);