#6576957

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573635
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	#60377	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2584 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)(3,8)(4,6)(5,7)(9,10)(11,18)(12,27)(13,20)(14,21)(15,33)(16,22)(17,23)(19,29)(24,40)(25,34)(26,36)(28,35)(30,39)(31,37)(32,38);(1,3,5,4)(2,6,7,8)(9,11,17,13)(10,18,23,20)(12,14,30,32)(15,26,16,28)(19,37,34,40)(21,39,38,27)(22,35,33,36)(24,29,31,25);(9,12)(10,19)(11,24)(13,31)(14,28)(15,25)(16,29)(17,30)(18,21)(20,38)(22,39)(23,34)(26,32)(27,33)(35,37)(36,40);(1,4,5,3)(2,6,7,8)(9,13,17,11)(10,20,23,18)(12,14,30,32)(15,28,16,26)(19,37,34,40)(21,39,38,27)(22,36,33,35)(24,29,31,25);(9,14,17,32)(10,21,23,38)(11,25,13,29)(12,28,30,26)(15,31,16,24)(18,34,20,19)(22,40,33,37)(27,35,39,36)

Polycyclic presentation:

2^9|b^2=fi,d^2=fi,e^2=i,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=dh,g^c=gi,h^c=hi,e^d=ehi,h^d=hi,h^e=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6484153);