#6621380

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574100
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	#61403	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1795 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fg,d^2=h,e^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,g^c=gi,e^d=eh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6528576);