#465403

order	512
nilpotency class	3
isoclinism	isoclinic class of #456510
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

880 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=fi,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,h^b=hi,d^c=de,e^c=ei,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, 372599);