#415880

order	512
nilpotency class	3
isoclinism	isoclinic class of #394271
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1578 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 323076);