#6446132

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438880
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1729 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=fg,d^2=f,f^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,h^c=hi,h^d=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6353328);