#119088

order	512
nilpotency class	3
isoclinism	isoclinic class of #10645
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

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	#1440
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

542 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=d,d^2=i,e^2=hi,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,d^b=di,e^b=eh,f^b=fh,g^b=ghi,e^c=ei,f^c=fhi,g^c=ghi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 26284);