#7562786

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
prime radical	#61347	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1656 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=gi,d^2=g,e^2=f,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, 7469982);