#497108

order	512
nilpotency class	4
isoclinism	isoclinic class of #483987
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#442
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂³	C₂×AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

473 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)(3,10)(4,8)(5,14)(6,9)(7,12)(11,16)(13,15)(18,24)(19,27,30,28)(20,22)(21,26)(25,29,32,31);(1,3)(2,7)(4,13)(5,11)(6,12)(8,16)(9,10)(14,15)(17,18,22,21,23,26,20,24)(19,28,29,32,30,27,31,25);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,10,16)(17,19,21,27,22,31,26,32,23,30,24,28,20,29,18,25);(1,5,6,4)(2,8,9,14)(3,11,12,13)(7,16,10,15)(17,18,20,24,23,26,22,21)(19,25,29,28,30,32,31,27)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=efh,d^2=e,e^2=i,f^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,h^a=hi,c^b=ce,d^b=di,f^b=fi,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404304);