#9759

order	256
nilpotency class	3
isoclinism	isoclinic class of #9758
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	#390
commutator	C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
prime radical	C₂³	#127
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

185 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)(3,9)(4,12)(5,7)(6,11)(8,13)(10,15)(14,16)(17,18)(19,23,28,38)(20,30)(21,35)(22,25)(24,33)(26,32)(27,41,31,45)(29,44,42,37)(34,46)(36,47,39,43)(40,48);(1,3,4,11,5,10,13,16)(2,6,8,9,7,14,12,15)(17,19)(18,23,25,38)(20,31)(21,29)(22,28)(24,39,40,36)(26,37,35,44)(27,34)(30,45,46,41)(32,42)(33,47)(43,48);(2,7)(3,10)(8,12)(11,16)(17,20,32,48,22,34,21,33)(18,24,35,46,25,40,26,30)(19,27,42,47,28,31,29,43)(23,36,44,45,38,39,37,41)

Polycyclic presentation:

2^8|a^2=f,b^2=df,c^2=eg,d^2=gh,e^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,c^b=cg,d^b=dh,f^b=fh

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#94563, #139074, #139075, …
maximal normal subgroup of	#94563, #139074, #139075, …
maximal quotient of	#102040, #102042, #102053, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 2747);