#8796

order	256
nilpotency class	3
isoclinism	isoclinic class of #8790
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

114 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:

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

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,f^b=fgh,f^c=fgh

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#135147, #135148, #135149, …
maximal normal subgroup of	#135147, #135148, #135149, …
maximal quotient of	#99276, #99279, #99281, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 1784);