#94575

order	512
nilpotency class	3
isoclinism	isoclinic class of #99
automorphism	?
homology	Abelianization (H₁) = C₂×C₆₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₁₆	C₂²⋊C₄
commutator	C₂²	C₂×C₆₄
Frattini, cosocle	C₂²×C₃₂	C₂²
socle	C₂²	C₂²⋊C₃₂
prime radical	C₂³	C₂×C₃₂
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂²×C₃₂ ◁ G
LECS	G ▷ C₂²×C₃₂ ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

84 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 2 generators on 72 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1771);