#410098

order	512
nilpotency class	3
isoclinism	isoclinic class of #395246
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1092 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=h,c^2=g,d^2=ei,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,g^b=gi,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 317294);