#6357416

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357390
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#127
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	C₂×#3147	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2169 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=f,c^2=gi,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,d^b=di,e^b=eh

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, 6264612);