#466449

order	512
nilpotency class	3
isoclinism	isoclinic class of #454426
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

970 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)(9,10,12,13)(11,15,16,14)(17,18)(19,27)(20,31)(21,25)(22,26)(23,32)(24,29)(28,30);(1,2,4,5)(3,6,7,8)(10,13)(14,15)(17,19,23,31,21,29,22,30)(18,24,32,28,25,27,26,20);(3,7)(6,8)(9,11)(10,14)(12,16)(13,15)(17,18)(19,28)(20,29)(21,25)(22,32)(23,26)(24,31)(27,30);(1,3,4,7)(2,6,5,8)(11,16)(14,15)(17,20)(18,19)(21,28)(22,27)(23,24)(25,29)(26,30)(31,32)

Polycyclic presentation:

2^9|a^2=f,b^2=efg,d^2=efi,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, 373645);