#465975

order	512
nilpotency class	3
isoclinism	isoclinic class of #456501
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

862 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)(3,6)(4,8)(5,7)(9,10)(11,21)(12,17)(13,19)(14,20)(15,18)(16,23)(22,36)(24,35)(25,39)(26,33)(27,34)(28,32)(29,31)(30,37)(38,40);(1,3,5,4)(2,6,7,8)(9,11,14,24)(10,16)(12,25,28,22)(13,23,15,29)(17,30)(18,21)(19,35)(20,31)(26,38,27,37)(32,40)(33,39)(34,36);(2,7)(6,8)(9,12,14,28)(10,17,20,32)(11,22,24,25)(13,26,15,27)(16,30,31,40)(18,34,19,33)(21,36,35,39)(23,37,29,38);(1,4,5,3)(2,8,7,6)(10,18)(12,26)(13,15)(16,21)(17,32)(19,20)(22,37)(23,29)(25,38)(27,28)(30,40)(31,35)

Polycyclic presentation:

2^9|b^2=fg,c^2=i,d^2=f,e^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,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, 373171);