#153579

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₂²×C₆₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

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

122 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

(1,2)(3,6)(4,5)(7,8)(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);(2,5)(6,8);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

2^9|a^2=e,c^2=d,e^2=f,f^2=g,g^2=h,h^2=i,b^a=bd

Conjugacy classes & character table

320 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60775);