#351290

order	512
nilpotency class	3
isoclinism	isoclinic class of #347638
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2856	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

703 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)(3,5)(4,6)(7,8)(10,12)(11,14)(15,16)(18,22)(19,25)(20,27)(23,30)(24,29)(28,31);(1,3,4,7)(2,5,6,8)(9,10,14,16,13,15,11,12)(17,18,20,22,21,24,27,29)(19,23,26,28,25,31,32,30);(2,6)(5,8)(9,11,13,14)(10,12,15,16);(9,12,11,15,13,16,14,10)(17,19,20,26,21,25,27,32)(18,23,22,28,24,31,29,30)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 258486);