#6447137

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438886
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#60425	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2050 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)(3,6)(4,8)(5,7)(9,10,15,16,14,20,13,19)(11,17,26,28,21,32,23,30)(12,18,27,29,24,34,25,33)(22,31,37,38,35,40,36,39);(1,3,5,4)(2,6,7,8)(10,16)(11,21)(12,24)(13,15)(17,30)(18,33)(19,20)(22,35)(28,32)(29,34)(31,39)(38,40);(1,4,5,3)(2,6,7,8)(9,11,13,23,14,21,15,26)(10,17,19,30,20,32,16,28)(12,22,25,36,24,35,27,37)(18,31,33,39,34,40,29,38);(9,12,13,25,14,24,15,27)(10,18,19,33,20,34,16,29)(11,22,23,36,21,35,26,37)(17,31,30,39,32,40,28,38);(11,21)(17,32)(22,35)(23,26)(28,30)(31,40)(36,37)(38,39)

Polycyclic presentation:

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

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, 6354333);