#345518

order	512
nilpotency class	3
isoclinism	isoclinic class of #333407
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

577 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 252714);