#505699

order	512
nilpotency class	4
isoclinism	isoclinic class of #500843
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

511 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)(10,16)(11,22)(12,17)(13,14)(18,33)(19,30)(20,29)(21,31)(23,37)(24,27)(25,26)(28,34)(35,40)(36,39);(1,3)(2,5)(4,7)(6,8)(9,10,17,30,14,20,26,31,15,21,25,29,13,19,12,16)(11,18,32,33,22,34,37,39,24,35,38,40,27,36,23,28);(3,7)(5,8)(9,11)(10,18)(12,23)(13,27)(14,22)(15,24)(16,28)(17,32)(19,36)(20,34)(21,35)(25,38)(26,37)(29,40)(30,33)(31,39);(9,10,17,30,14,20,26,31,15,21,25,29,13,19,12,16)(11,18,32,33,22,34,37,39,24,35,38,40,27,36,23,28)

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,f^c=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 412895);