#13702

order	256
nilpotency class	5
isoclinism	isoclinic class of #13700
automorphism	Aut(G) = D₄×#12067 (order 2048) · Inn(G) = #2215 · Out(G) = C₂×C₈
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#2215
commutator, Frattini, cosocle	C₂×C₁₆	C₂³
prime radical	#2221	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #131 ◁ #504 ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

92 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 64 points:

(2,9)(3,16)(4,20)(5,7)(6,21)(10,24)(11,38)(12,32)(13,44)(14,30)(15,33)(17,48)(18,51)(19,25)(22,36)(23,37)(26,27)(28,39)(29,31)(34,43)(35,52)(40,59)(41,57)(42,56)(45,58)(46,49)(50,60)(54,62)(55,61)(63,64);(1,2,10,34,6,13,23,32,5,12,36,44,26,43,4,9,8,15,20,31,27,45,22,30,7,14,37,58,21,29,24,33)(3,11,35,59,18,40,46,38,25,42,48,62,53,63,17,28,19,41,49,55,51,61,52,57,16,39,60,64,47,54,50,56);(1,3)(2,11)(4,17)(5,25)(6,18)(7,16)(8,19)(9,28)(10,35)(12,42)(13,40)(14,39)(15,41)(20,49)(21,47)(22,52)(23,46)(24,50)(26,53)(27,51)(29,54)(30,57)(31,55)(32,38)(33,56)(34,59)(36,48)(37,60)(43,63)(44,62)(45,61)(58,64)

Polycyclic presentation:

2^8|b^2=dfh,d^2=fg,e^2=h,f^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eh,f^a=fg,g^a=gh,e^c=eh

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#13702, #512951, #512952, …
maximal normal subgroup of	C₂×#13702, #512951, #512952, …
maximal quotient of	#151639, #151642, #151648, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 6690);