#326924

order	512
nilpotency class	3
isoclinism	isoclinic class of #326869
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

843 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=hi,d^2=hi,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=dh,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=ghi,d^c=de,f^c=fh,g^c=ghi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 234120);