#6585823

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574070
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	#28504	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1896 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fg,d^2=f,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6493019);