#6437707

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	C₂²×#578	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4556 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,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, 6344903);