#461004

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

721 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)(3,5)(4,6)(7,8)(10,12)(11,15)(17,18,23,31,21,20,22,25)(19,26,30,27,28,24,29,32);(1,3,4,7)(2,5,6,8)(9,10,13,12)(11,16,15,14)(17,19,22,29,21,28,23,30)(18,24,25,27,20,26,31,32);(2,6)(5,8)(9,11,13,15)(10,14,12,16)(17,20,21,18)(19,27,28,32)(22,25,23,31)(24,30,26,29);(9,12,13,10)(11,14,15,16)(17,18)(19,24)(20,21)(22,25)(23,31)(26,28)(27,29)(30,32)

Polycyclic presentation:

2^9|a^2=eg,b^2=f,c^2=e,d^2=ei,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^c=de,f^c=fi,g^c=gi,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, 368200);