#7560680

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519063
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	#61170	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1734 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)(11,17)(12,24)(13,22)(14,19)(15,32)(16,20)(18,28)(21,35)(23,30)(25,34)(26,37)(27,36)(29,38)(31,33)(39,40);(2,5)(6,8)(9,11)(10,17)(12,25)(13,19)(14,22)(15,33)(16,23)(18,35)(20,30)(21,28)(24,34)(26,36)(27,37)(29,39)(31,32)(38,40);(1,3,4,7)(2,6,5,8)(9,12,15,28,16,29,14,27)(10,18,19,24,20,36,32,38)(11,21,22,25,23,37,33,39)(13,26,17,34,31,35,30,40);(9,13,15,17,16,31,14,30)(10,11,19,22,20,23,32,33)(12,26,28,34,29,35,27,40)(18,21,24,25,36,37,38,39);(2,5)(6,8)(9,14,16,15)(10,19,20,32)(11,22,23,33)(12,27,29,28)(13,30,31,17)(18,24,36,38)(21,25,37,39)(26,40,35,34)

Polycyclic presentation:

2^9|c^2=fg,d^2=g,e^2=i,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,h^d=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7467876);