#7562436

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519075
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2130 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|c^2=g,d^2=g,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cgi,d^b=dh,g^b=gi,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, 7469632);