#13721

order	256
nilpotency class	5
isoclinism	isoclinic class of #13694
automorphism	Aut(G) order 8192
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂³
prime radical	C₂×D₁₆	C₂²
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

90 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 72 points:

(1,2)(3,6)(4,5)(7,8)(9,10)(11,25)(12,18)(13,35)(14,39)(15,41)(16,44)(17,24)(19,30)(20,26)(21,38)(22,36)(23,42)(27,49)(28,60)(29,50)(31,55)(32,54)(33,66)(34,48)(37,52)(40,51)(43,70)(45,64)(46,63)(47,65)(53,69)(56,61)(57,68)(58,67)(59,72)(62,71);(2,5)(6,8)(9,11,26,32,15,28,57,65,42,61,40,59,69,64,38,34,17,30,14,27,43,62,37,33,16,29,58,63,36,31,13,12)(10,18,35,55,22,46,67,50,44,66,52,71,70,49,39,19,24,48,21,45,53,72,51,56,23,47,68,60,41,54,20,25);(1,3,4,7)(2,6,5,8)(9,12,13,31,36,63,58,29,16,33,37,62,43,27,14,30,17,34,38,64,69,59,40,61,42,65,57,28,15,32,26,11)(10,19,20,49,41,71,68,66,23,50,51,46,53,55,21,18,24,25,39,54,70,60,52,47,44,56,67,72,22,45,35,48)

Polycyclic presentation:

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

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#151578, #151579, C₂×#13721, …
maximal normal subgroup of	#151578, #151579, C₂×#13721, …
maximal quotient of	#152385, #152830, #152831, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 6709);