#463559

order	512
nilpotency class	3
isoclinism	isoclinic class of #454426
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

792 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:

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

Polycyclic presentation:

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

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, 370755);