#238172

order	512
nilpotency class	3
isoclinism	isoclinic class of #234044
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

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

820 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 145368);