#465583

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

725 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=eg,c^2=ef,d^2=ei,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^c=de,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, 372779);