#6454849

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437963
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3047 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6362045);