#7565687

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	#61092	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1913 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fg,d^2=fg,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, 7472883);