#419776

order	512
nilpotency class	3
isoclinism	isoclinic class of #417692
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1617 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=i,d^2=ei,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,g^b=gi,f^c=fi,g^c=gi,h^c=hi,f^d=fi,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, 326972);