#6578408

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	#25802	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1958 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)(11,23,25,17)(12,19,26,33)(13,20,29,37)(15,18,16,22)(24,39,28,32)(27,35,40,36)(30,34,31,38);(2,6)(7,8)(9,11)(10,17)(12,16)(13,27)(14,25)(15,26)(18,19)(20,36)(21,23)(22,33)(24,30)(28,31)(29,40)(32,34)(35,37)(38,39);(10,18)(11,12)(15,16)(19,33)(20,34)(21,22)(24,27)(25,26)(28,40)(30,31)(35,36)(37,38);(1,3,5,4)(2,7,6,8)(9,12)(10,19)(11,15)(13,28)(14,26)(16,25)(17,22)(18,23)(20,32)(21,33)(24,29)(27,30)(31,40)(34,35)(36,38)(37,39);(1,4,5,3)(2,8,6,7)(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|a^2=f,d^2=fi,e^2=fi,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, 6485604);