#411180

order	512
nilpotency class	3
isoclinism	isoclinic class of #395169
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#127	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1098 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=fg,b^2=ghi,c^2=g,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=di,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 318376);