#434763

order	512
nilpotency class	3
isoclinism	isoclinic class of #422715
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
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

992 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)(10,15)(11,19)(13,25)(16,28)(17,29)(18,30)(22,34)(23,27)(31,39)(32,33)(36,40)(37,38);(1,3,4,8)(2,5,6,7)(9,10,13,15,14,18,25,30)(11,16,21,32,19,33,26,28)(12,22,23,34,24,29,27,17)(20,36,37,40,35,39,38,31);(2,6)(5,7)(9,11,13,26,14,19,25,21)(10,16,30,32,18,33,15,28)(12,20,27,37,24,35,23,38)(17,31,22,39,34,40,29,36);(3,8)(5,7)(9,12,14,24)(10,17,18,34)(11,20,19,35)(13,27,25,23)(15,29,30,22)(16,31,33,40)(21,38,26,37)(28,36,32,39)

Polycyclic presentation:

2^9|a^2=fg,b^2=e,c^2=ef,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,f^b=fi,g^b=gi,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, 341959);