#371878

order	512
nilpotency class	3
isoclinism	isoclinic class of #371877
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14082
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1072 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=ef,d^2=gi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,h^b=hi,e^c=ei,f^c=fi,e^d=ei,f^d=fi,h^d=hi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 279074);