#93970

order	512
nilpotency class	4
isoclinism	isoclinic class of #1366
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₆ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#333
commutator	C₄²	C₂×C₁₆
Frattini, cosocle	C₂×C₄×C₁₆	C₂²
socle	C₂	#7128
prime radical	C₂³	#349
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₁₆ ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

132 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=f,c^2=fi,d^2=g,e^2=i,g^2=h,h^2=i,b^a=bc,c^a=ce,e^a=ei,f^a=fi,c^b=cf,d^b=def,e^b=ei,g^b=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1166);