#6357303

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357300
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	#60639	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2200 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6264499);