#6362757

order	512
nilpotency class	3
isoclinism	isoclinic class of #6362750
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	C₄²	C₂⁵
prime radical	#61901	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1742 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)(3,5)(4,6)(7,8)(9,10,14,17)(11,19,20,35)(12,23,26,16)(13,27,30,18)(15,28,31,22)(21,33,38,37)(24,32,39,40)(25,34,29,36);(9,11,14,20)(10,15,17,31)(12,24,26,39)(13,28,30,22)(16,33,23,37)(18,35,27,19)(21,29,38,25)(32,36,40,34);(2,5)(3,8)(6,7)(11,20)(15,31)(19,35)(21,38)(22,28)(24,39)(32,40)(33,37);(9,12,14,26)(10,16,17,23)(11,21,20,38)(13,29,30,25)(15,32,31,40)(18,36,27,34)(19,37,35,33)(22,39,28,24);(2,6)(5,7)(10,17)(15,31)(16,23)(18,27)(19,35)(32,40)(33,37)(34,36)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6269953);