#6520025

order	512
nilpotency class	3
isoclinism	isoclinic class of #6502819
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	#3444	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1690 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:

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

Polycyclic presentation:

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

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, 6427221);