#6466886

order	512
nilpotency class	3
isoclinism	isoclinic class of #6443048
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2086 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6374082);