#6456433

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	#60559	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1816 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)(9,10,16,33,15,21,14,20)(11,22,32,17,26,40,25,36)(12,27,31,38,29,18,13,19)(23,39,24,37,28,35,30,34);(9,11)(10,17)(12,28)(13,30)(14,32)(15,26)(16,25)(18,37)(19,39)(20,40)(21,36)(22,33)(23,29)(24,31)(27,34)(35,38);(2,5)(6,8)(9,12,16,31,15,29,14,13)(10,18,33,19,21,27,20,38)(11,23,32,24,26,28,25,30)(17,34,40,39,36,37,22,35);(9,13,14,29,15,31,16,12)(10,19,20,18,21,38,33,27)(11,24,25,23,26,30,32,28)(17,35,22,37,36,39,40,34);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

2^9|a^2=fi,c^2=fi,d^2=f,e^2=gi,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, 6363629);