#6684984

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682658
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#61090	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1899 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=fg,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6592180);