#505703

order	512
nilpotency class	4
isoclinism	isoclinic class of #500848
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#525	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

493 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,d^2=eh,e^2=hi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,h^a=hi,c^b=cg,d^b=di,f^c=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, 412899);