#476169

order	512
nilpotency class	3
isoclinism	isoclinic class of #476092
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×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1349 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)(5,6)(7,8)(9,10,13,18,14,19,29,16)(11,20,22,32,23,35,28,15)(12,24,26,34,27,36,30,17)(21,31,38,40,25,33,39,37);(5,6)(7,8)(9,11,14,23)(10,15,19,32)(12,25,27,21)(13,28,29,22)(16,20,18,35)(17,33,34,31)(24,40,36,37)(26,38,30,39);(6,8)(10,16)(12,26)(13,29)(15,20)(17,34)(18,19)(21,38)(22,28)(25,39)(27,30)(32,35)(37,40);(2,4)(6,8)(9,12)(10,17)(11,21)(13,30)(14,27)(15,31)(16,24)(18,36)(19,34)(20,37)(22,39)(23,25)(26,29)(28,38)(32,33)(35,40)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 383365);