#458325

order	512
nilpotency class	3
isoclinism	isoclinic class of #454372
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2675
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

758 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)(3,6)(4,8)(5,7)(9,10)(11,20)(12,25)(13,28)(14,30)(15,19)(16,23)(17,26)(18,29)(21,31)(22,39)(24,36)(27,34)(32,38)(33,40)(35,37);(1,3,5,4)(2,6,7,8)(9,11,14,23,15,24,29,40)(10,16,18,20,19,33,30,36)(12,21,26,37,27,38,13,22)(17,32,25,35,28,31,34,39);(1,4,5,3)(2,6,7,8)(9,12,15,27)(10,17,19,28)(11,21,24,38)(13,29,26,14)(16,31,33,32)(18,34,30,25)(20,35,36,39)(22,40,37,23);(9,13,14,12,15,26,29,27)(10,17,18,25,19,28,30,34)(11,22,23,21,24,37,40,38)(16,32,20,35,33,31,36,39)

Polycyclic presentation:

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

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, 365521);