#434245

order	512
nilpotency class	3
isoclinism	isoclinic class of #417784
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1282 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:

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

Polycyclic presentation:

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

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, 341441);