#8798

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:

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

Polycyclic presentation:

2^8|a^2=f,b^2=e,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	#146102, #146103, #146104, …
maximal normal subgroup of	#146102, #146103, #146104, …
maximal quotient of	#99282, #99283, #99286, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 1786);