#94561

order	512
nilpotency class	6
isoclinism	isoclinic class of #94560
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#7352
commutator	C₄×C₈	C₂×C₈
Frattini, cosocle	#1837	C₂²
prime radical	C₂×#493	C₄
LCS	G ▷ C₄×C₈ ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1837 ◁ G
LECS	G ▷ #1837 ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

165 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=d,c^2=gi,d^2=f,e^2=h,f^2=i,g^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gh,h^a=hi,c^b=cg,d^b=degh,e^b=eh,f^b=fi,g^b=gi,d^c=dghi,e^d=ehi,g^d=gi

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1757);