#461103

order	512
nilpotency class	3
isoclinism	isoclinic class of #455572
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

901 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=f,e^2=i,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,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,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, 368299);