#6456427

order	512
nilpotency class	3
isoclinism	isoclinic class of #25140
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×#121
prime radical	#3444	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1808 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)(4,5)(7,8)(9,10,14,19,15,20,26,33)(11,21,24,32,25,36,29,16)(12,17,27,34,13,18,28,35)(22,37,39,31,23,38,40,30);(9,11,15,25)(10,16,20,32)(12,23,13,22)(14,29,26,24)(17,31,18,30)(19,36,33,21)(27,39,28,40)(34,37,35,38);(2,5)(6,8)(9,12,14,27,15,13,26,28)(10,17,19,34,20,18,33,35)(11,22,24,39,25,23,29,40)(16,30,21,37,32,31,36,38);(9,13,14,28,15,12,26,27)(10,18,19,35,20,17,33,34)(11,23,24,40,25,22,29,39)(16,31,21,38,32,30,36,37);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6363623);