#120741

order	512
nilpotency class	3
isoclinism	isoclinic class of #10637
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#397
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1442
prime radical	C₂⁴	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

455 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=d,d^2=h,f^2=hi,g^2=hi,b^a=bd,c^a=ce,d^a=dh,c^b=cf,d^b=dh,f^b=fi,e^c=eh,f^c=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 27937);