#6574975

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

1991 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)(12,19)(13,20)(14,21)(15,18)(16,22)(17,25)(24,39)(26,33)(27,35)(28,32)(29,37)(30,34)(31,38)(36,40);(2,5)(6,8)(9,11,14,25)(10,17,21,23)(12,15,26,16)(13,27,29,40)(18,19,22,33)(20,36,37,35)(24,31,28,30)(32,38,39,34);(1,3,4,7)(2,6,5,8)(10,18)(11,12)(15,16)(19,33)(20,34)(21,22)(24,27)(25,26)(28,40)(30,31)(35,36)(37,38);(9,12,14,26)(10,19,21,33)(11,16,25,15)(13,28,29,24)(17,18,23,22)(20,32,37,39)(27,31,40,30)(34,35,38,36);(9,13)(10,20)(11,24)(12,27)(14,29)(15,30)(16,31)(17,32)(18,34)(19,35)(21,37)(22,38)(23,39)(25,28)(26,40)(33,36)

Polycyclic presentation:

2^9|b^2=i,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, 6482171);