#329418

order	512
nilpotency class	3
isoclinism	isoclinic class of #329395
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	C₂²×#41	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

746 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)(3,6)(4,5)(7,8)(9,10,13,17,14,18,12,16)(11,19,20,23,21,24,22,15)(25,26,28,32,29,33,37,31)(27,30,34,38,36,40,35,39);(2,5)(6,8)(25,27)(26,30)(28,34)(29,36)(31,39)(32,38)(33,40)(35,37);(1,3)(2,6)(4,7)(5,8)(9,11,14,21)(10,15,18,23)(12,20,13,22)(16,19,17,24)(26,31)(27,34)(28,37)(32,33)(35,36)(38,39);(3,7)(6,8)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,19,23,24)

Polycyclic presentation:

2^9|a^2=f,c^2=hi,d^2=hi,f^2=h,g^2=i,b^a=be,c^a=cf,c^b=cg,g^b=gi,d^c=de,f^c=fh,g^c=gi

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, 236614);