#436910

order	512
nilpotency class	3
isoclinism	isoclinic class of #422131
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
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

959 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:

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

Polycyclic presentation:

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

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, 344106);