#454562

order	512
nilpotency class	3
isoclinism	isoclinic class of #454029
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1222 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 361758);