#8791

order	256
nilpotency class	3
isoclinism	isoclinic class of #8790
automorphism	Aut(G) order 16384
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×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	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

131 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^8|a^2=f,b^2=eg,c^2=d,d^2=h,e^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	#94580, #135119, #135120, …
maximal normal subgroup of	#94580, #135119, #135120, …
maximal quotient of	#99277, #99281, #99284, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 1779);