#8799

order	256
nilpotency class	3
isoclinism	isoclinic class of #8794
automorphism	Aut(G) order 8192
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#382
commutator	C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
prime radical	C₂³	#125
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

110 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#8799, #191259, #191261, …
maximal normal subgroup of	C₂×#8799, #191259, #191261, …
maximal quotient of	#99282, #99284, #99285, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 1787);