#6466876

order	512
nilpotency class	3
isoclinism	isoclinic class of #24802
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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	C₂×#3147	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2124 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)(9,10,15,22)(11,23,27,35)(12,28,31,18)(13,19,30,37)(14,20,16,21)(17,32,36,26)(24,34,40,39)(25,33,29,38);(2,5)(6,7)(9,11,16,32,15,27,14,26)(10,17,20,35,22,36,21,23)(12,29,13,24,31,25,30,40)(18,34,19,38,28,39,37,33);(1,3,4,8)(2,6,5,7)(9,12,14,30,15,31,16,13)(10,18,20,19,22,28,21,37)(11,24,32,25,27,40,26,29)(17,33,23,39,36,38,35,34);(9,13,16,31,15,30,14,12)(10,19,21,18,22,37,20,28)(11,25,26,24,27,29,32,40)(17,34,35,38,36,39,23,33);(9,14,15,16)(10,20,22,21)(11,26,27,32)(12,30,31,13)(17,35,36,23)(18,19,28,37)(24,29,40,25)(33,34,38,39)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6374072);