#211977

order	512
nilpotency class	3
isoclinism	isoclinic class of #19186
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₄×#121
prime radical	C₂×#41	#41
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

550 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,d^2=gh,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=ch,d^c=dh,f^c=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 119173);