#6442679

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×#3485	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4682 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)(9,10,15,20)(11,21,25,31)(12,17,27,34)(13,19,14,18)(16,24,32,23)(22,33,37,40)(26,30,38,39)(28,36,29,35);(2,4)(9,11,15,25)(10,16,20,32)(12,26,27,38)(13,24,14,23)(17,33,34,40)(18,31,19,21)(22,28,37,29)(30,35,39,36);(5,6,7,8)(9,12)(10,17)(11,22)(13,29)(14,28)(15,27)(16,30)(18,36)(19,35)(20,34)(21,33)(23,26)(24,38)(25,37)(31,40)(32,39);(11,23)(13,14)(16,21)(18,19)(22,26)(24,25)(28,29)(30,33)(31,32)(35,36)(37,38)(39,40);(12,27)(17,34)(22,37)(26,38)(28,29)(30,39)(33,40)(35,36)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=g,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, 6349875);