#435545

order	512
nilpotency class	3
isoclinism	isoclinic class of #418818
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2684
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15529
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

749 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:

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

Polycyclic presentation:

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

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, 342741);