#6466568

order	512
nilpotency class	3
isoclinism	isoclinic class of #6443050
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#3449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1858 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=fg,c^2=ghi,d^2=h,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,h^b=hi,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6373764);