#8797

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:

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

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

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#135055, #135056, #135057, …
maximal normal subgroup of	#135055, #135056, #135057, …
maximal quotient of	#99277, #99278, #99280, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 1785);