#6437546

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437537
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹¹

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 ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

6134 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)(9,10)(11,21)(12,26)(13,18)(14,19)(15,20)(16,24)(17,27)(22,36)(23,32)(25,31)(28,34)(29,33)(30,39)(35,38)(37,40);(2,4)(5,6)(7,8)(9,11,15,25)(10,16,20,32)(12,22,29,39)(13,23,14,24)(17,30,34,36)(18,21,19,31)(26,40,33,35)(27,37,28,38);(6,8)(9,12)(10,17)(11,22)(13,27)(14,28)(15,29)(16,30)(18,33)(19,26)(20,34)(21,35)(23,37)(24,38)(25,39)(31,40)(32,36);(6,8)(9,13,15,14)(10,18,20,19)(11,23,25,24)(12,27,29,28)(16,21,32,31)(17,33,34,26)(22,37,39,38)(30,35,36,40);(9,13,15,14)(10,19,20,18)(11,24,25,23)(12,27,29,28)(16,21,32,31)(17,26,34,33)(22,38,39,37)(30,35,36,40)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314786, #72314796, #72314820, …
maximal normal subgroup of	#72314786, #72314796, #72314820, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6344742);