#459706

order	512
nilpotency class	3
isoclinism	isoclinic class of #453948
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

966 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=f,d^2=efi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,g^b=gi,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, 366902);