#439026

order	512
nilpotency class	3
isoclinism	isoclinic class of #418500
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1403 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 346222);