#453862

order	512
nilpotency class	3
isoclinism	isoclinic class of #421159
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

727 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

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, 361058);