#6574974

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	#61062	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1987 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)(3,6)(4,5)(7,8)(9,10)(11,23)(12,27)(13,18)(14,20)(15,21)(16,33)(17,22)(19,25)(24,38)(26,36)(28,37)(29,40)(30,35)(31,39)(32,34);(2,5)(6,8)(9,11)(10,18)(12,14)(13,15)(16,26)(17,28)(19,38)(20,40)(21,23)(22,36)(24,30)(25,31)(27,34)(29,32)(33,37)(35,39);(1,3,4,7)(2,6,5,8)(9,12)(10,19)(11,24)(13,31)(14,28)(15,29)(16,30)(17,25)(18,34)(20,23)(21,35)(22,40)(26,32)(27,33)(36,38)(37,39);(9,13,15,11)(10,18,21,23)(12,14,29,32)(16,28,17,26)(19,39,35,38)(20,40,34,27)(22,36,33,37)(24,25,31,30);(9,14)(10,20)(11,25)(12,28)(13,30)(15,32)(16,31)(17,24)(18,35)(19,23)(21,34)(22,38)(26,29)(27,37)(33,39)(36,40)

Polycyclic presentation:

2^9|c^2=fi,d^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,h^b=hi,d^c=dh,g^c=gi,h^c=hi,e^d=eh,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, 6482170);