#6431908

order	512
nilpotency class	3
isoclinism	isoclinic class of #6431707
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4714 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=i,c^2=f,d^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,h^a=hi,c^b=ch,d^b=di,g^b=gi,h^b=hi,g^c=gi,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6339104);