#7566055

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519083
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	#61096	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1897 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)(11,21,24,16)(12,13,27,29)(15,31,28,20)(17,35,30,18)(22,39,33,36)(23,26,38,40)(25,34,32,37);(2,6)(7,8)(9,11,14,24)(10,16,19,21)(12,26,27,40)(13,23,29,38)(15,32,28,25)(17,36,30,39)(18,22,35,33)(20,37,31,34);(1,3,5,4)(2,7,6,8)(9,12,15,18,14,27,28,35)(10,17,20,13,19,30,31,29)(11,22,25,26,24,33,32,40)(16,23,34,36,21,38,37,39);(1,3,5,4)(2,7,6,8)(9,13,28,17,14,29,15,30)(10,18,31,12,19,35,20,27)(11,23,32,39,24,38,25,36)(16,33,37,26,21,22,34,40);(1,4,5,3)(2,7,6,8)(10,19)(12,27)(16,21)(18,35)(20,31)(22,33)(26,40)(34,37)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fgi,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=cg,d^b=dh,g^b=gi,e^c=ei,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, 7473251);