#6437708

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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4620 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)(5,6)(7,8)(10,15)(16,30)(17,31)(18,27)(19,33)(23,34)(32,39)(35,40);(2,4)(9,10,13,15)(11,19,21,33)(12,23,24,34)(14,27,26,18)(16,28,30,22)(17,29,31,25)(20,32,36,39)(35,38,40,37);(6,8)(9,11)(10,16)(12,20)(13,21)(14,28)(15,30)(17,32)(18,33)(19,27)(22,26)(23,35)(24,36)(25,38)(29,37)(31,39)(34,40);(9,12,13,24)(10,17,15,31)(11,20,21,36)(14,29,26,25)(16,32,30,39)(18,34,27,23)(19,35,33,40)(22,38,28,37);(11,21)(16,30)(19,33)(20,36)(22,28)(32,39)(35,40)(37,38)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,d^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,h^b=hi,e^c=ei,h^c=hi,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, 6344904);