#6437711

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

4594 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)(11,20)(12,25)(13,18)(14,28)(15,19)(16,23)(17,26)(21,30)(22,31)(24,34)(27,32)(29,33)(35,36)(37,40)(38,39);(6,8)(9,11,15,24)(10,16,19,31)(12,21,27,37)(13,23,14,22)(17,30,33,40)(18,34,28,20)(25,35,32,39)(26,38,29,36);(9,12)(10,17)(11,21)(13,26)(14,29)(15,27)(16,30)(18,32)(19,33)(20,35)(22,36)(23,38)(24,37)(25,28)(31,40)(34,39);(9,13,15,14)(10,18,19,28)(11,22,24,23)(12,26,27,29)(16,20,31,34)(17,32,33,25)(21,36,37,38)(30,35,40,39)

Polycyclic presentation:

2^9|a^2=f,c^2=i,e^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, 6344907);