#7518278

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514160
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	#60432	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1875 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)(3,7)(4,5)(6,8)(9,10)(11,16)(12,23)(13,28)(14,30)(15,20)(17,26)(18,29)(19,32)(21,38)(22,36)(24,40)(25,34)(27,37)(31,35)(33,39);(2,5)(6,7)(9,11,15,13)(10,16,20,28)(12,24,27,39)(14,29,32,22)(17,34,35,38)(18,19,36,30)(21,26,25,31)(23,40,37,33);(9,12)(10,17)(11,21)(13,25)(14,31)(15,27)(16,33)(18,34)(19,37)(20,35)(22,24)(23,30)(26,32)(28,40)(29,39)(36,38);(9,13,15,11)(10,18,20,36)(12,25,27,21)(14,22,32,29)(16,30,28,19)(17,34,35,38)(23,40,37,33)(24,26,39,31);(1,3,4,8)(2,6,5,7)

Polycyclic presentation:

2^9|b^2=i,d^2=i,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,g^c=gi,h^c=hi,g^d=gi

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, 7425474);