#477535

order	512
nilpotency class	3
isoclinism	isoclinic class of #476112
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1013 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:

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

Polycyclic presentation:

2^9|b^2=f,c^2=f,d^2=i,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,d^b=def,e^b=ei,d^c=de,e^c=ei,h^c=hi,e^d=ei,h^d=hi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 384731);