#351627

order	512
nilpotency class	3
isoclinism	isoclinic class of #347788
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

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

561 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,5)(4,6)(7,8)(9,10,13,17)(11,14,18,21)(12,15,20,16)(19,22,24,23)(26,31)(27,29)(28,36)(32,34)(33,37)(35,38);(1,3,4,7)(2,5,6,8)(11,18)(14,21)(19,24)(22,23)(25,26,27,32,30,34,29,31)(28,37,35,39,38,33,36,40);(2,6)(5,8)(9,11,13,18)(10,14,17,21)(12,19,20,24)(15,22,16,23)(25,27,30,29)(26,32,34,31)(28,36,38,35)(33,39,37,40);(10,15)(12,20)(14,22)(16,17)(19,24)(21,23)(25,28,29,36,30,38,27,35)(26,33,31,39,34,37,32,40)

Polycyclic presentation:

2^9|a^2=hi,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,c^b=chi,d^b=dh,d^c=dh,g^d=ghi

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, 258823);