#334822

order	512
nilpotency class	3
isoclinism	isoclinic class of #334695
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2669
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3205	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

660 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 242018);