#7560719

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	#61168	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1732 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)(4,5)(7,8)(9,10)(11,16)(12,23)(13,26)(14,31)(15,20)(17,22)(18,29)(19,32)(21,37)(24,33)(25,34)(27,38)(28,35)(30,40)(36,39);(2,5)(6,8)(9,11,15,22)(10,16,20,17)(12,14,25,32)(13,27,30,21)(18,33,36,35)(19,23,31,34)(24,39,28,29)(26,38,40,37);(1,3,4,7)(2,6,5,8)(9,12,14,22,15,25,32,11)(10,17,19,23,20,16,31,34)(13,28,29,21,30,24,39,27)(18,33,26,37,36,35,40,38);(9,13,14,29,15,30,32,39)(10,18,19,26,20,36,31,40)(11,21,12,24,22,27,25,28)(16,33,34,37,17,35,23,38);(2,5)(6,8)

Polycyclic presentation:

2^9|b^2=i,c^2=fg,d^2=g,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,d^c=di,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, 7467915);