#6442682

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
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	C₂×#3487	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4638 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)(6,8)(10,16)(13,14)(17,21)(18,26)(19,20)(24,25)(27,28)(30,35)(31,32)(33,34)(37,38)(39,40);(2,4)(9,10,15,20)(11,21,23,32)(12,26,29,33)(13,19,14,16)(17,24,31,25)(18,27,34,28)(22,30,36,39)(35,37,40,38);(5,6,7,8)(9,11)(10,17)(12,22)(13,25)(14,24)(15,23)(16,21)(18,30)(19,32)(20,31)(26,35)(27,38)(28,37)(29,36)(33,40)(34,39);(9,12,15,29)(10,18,20,34)(11,22,23,36)(13,28,14,27)(16,26,19,33)(17,30,31,39)(21,35,32,40)(24,38,25,37);(11,23)(17,31)(21,32)(22,36)(24,25)(30,39)(35,40)(37,38)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6349878);