#94577

order	512
nilpotency class	4
isoclinism	isoclinic class of #353
automorphism	?
homology	Abelianization (H₁) = C₂×C₃₂ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₁₆	#99
commutator	C₂×C₄	C₂×C₃₂
Frattini, cosocle	C₂×M₆₄	C₂²
socle	C₂	#7362
prime radical	C₂³	M₆₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ C₂×M₆₄ ◁ G
LECS	G ▷ C₂×M₆₄ ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

78 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=i,d^2=f,f^2=g,g^2=h,h^2=i,b^a=bc,c^a=ce,e^a=ei,d^b=dei,d^c=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1773);