#489624

order	512
nilpotency class	4
isoclinism	isoclinic class of #489355
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#33811	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

689 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=i,c^2=hi,d^2=ghi,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fh,g^a=gi,d^b=dg,e^b=eg,f^b=fh,g^b=gi,e^c=eh,f^c=fh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396820);