#129198

order	512
nilpotency class	3
isoclinism	isoclinic class of #447
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄×C₈	C₂×D₄
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₈×C₄⋊C₄
prime radical	C₂³	C₈×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

240 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 32 points:

(2,7,9,13)(3,10)(8,15,16,14)(11,12)(17,18,20,24,21,19,26,27,22,25,29,32,30,28,23,31);(1,2,5,7,6,9,4,13)(3,8,12,14,10,16,11,15);(1,3,4,11,6,10,5,12)(2,8,13,15,9,16,7,14)(17,19,20,27,21,25,26,32,22,28,29,31,30,18,23,24)

Polycyclic presentation:

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

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36394);