#7563460

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	C₂×#41	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1592 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=g,d^2=g,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,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, 7470656);