#7545944

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519083
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#535	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1645 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fgi,c^2=g,e^2=fi,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,h^b=hi,e^c=ei,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7453140);