#7518281

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	#60430	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1899 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)(11,20)(12,24)(13,18)(14,17)(15,22)(16,27)(19,28)(21,34)(23,33)(25,31)(26,36)(29,32)(30,35)(37,39)(38,40);(2,5)(6,7)(9,11,13,22)(10,15,18,20)(12,25,26,39)(14,29,28,23)(16,34,35,40)(17,33,19,32)(21,27,38,30)(24,37,36,31);(9,12)(10,16)(11,21)(13,26)(14,30)(15,31)(17,24)(18,35)(19,36)(20,37)(22,38)(23,25)(27,28)(29,39)(32,40)(33,34);(10,17)(14,28)(15,32)(16,24)(18,19)(20,33)(23,29)(25,39)(27,30)(31,40)(34,37)(35,36);(1,3,4,8)(2,6,5,7)(10,18)(15,20)(16,35)(17,19)(24,36)(31,37)(32,33)(34,40)

Polycyclic presentation:

2^9|b^2=i,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,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, 7425477);