#6359951

order	512
nilpotency class	3
isoclinism	isoclinic class of #6359920
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	#61880	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1814 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)(3,6)(4,7)(5,8)(9,10)(11,20)(12,17)(13,18)(14,28)(15,19)(16,23)(21,31)(22,37)(24,32)(25,33)(26,40)(27,35)(29,36)(30,39)(34,38);(1,3,5,4)(2,6,8,7)(9,11,15,24)(10,16,19,31)(12,23,14,21)(13,26,27,39)(17,32,28,20)(18,34,35,37)(22,29,38,25)(30,33,40,36);(2,7)(3,4)(6,8)(9,12,15,14)(10,17,19,28)(11,21,24,23)(13,25,27,29)(16,20,31,32)(18,33,35,36)(22,26,38,39)(30,37,40,34);(9,13,15,27)(10,18,19,35)(11,22,24,38)(12,25,14,29)(16,30,31,40)(17,33,28,36)(20,37,32,34)(21,26,23,39);(10,19)(16,31)(17,28)(18,35)(20,32)(30,40)(33,36)(34,37)

Polycyclic presentation:

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

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, 6267147);