#6578407

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	#25809	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1929 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=fi,e^2=f,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, 6485603);