#434169

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	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

967 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)(3,5,7,8)(10,16)(11,20)(12,24)(13,14)(17,30)(18,25)(19,28)(21,34)(22,23)(31,39)(32,38)(33,36);(1,3)(2,5)(4,7)(6,8)(9,10)(11,17)(12,25)(13,28)(14,16)(15,19)(18,26)(20,29)(21,35)(22,37)(23,30)(24,38)(27,32)(31,34)(33,39)(36,40);(3,7)(5,8)(9,11,14,20,15,23,13,22)(10,17,16,29,19,30,28,37)(12,21,27,33,24,36,26,34)(18,31,25,35,32,39,38,40);(9,12,13,26,15,24,14,27)(10,18,16,25,19,32,28,38)(11,21,22,34,23,36,20,33)(17,31,29,35,30,39,37,40)

Polycyclic presentation:

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

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, 341365);