#6439781

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437602
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2096 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)(11,17)(12,27)(13,19)(14,20)(15,21)(16,22)(18,30)(23,36)(24,34)(25,37)(26,35)(28,40)(29,39)(31,38)(32,33);(2,5)(6,8)(9,11,16,26)(10,17,22,35)(12,28,31,32)(13,30,23,29)(14,24,15,25)(18,36,39,19)(20,34,21,37)(27,40,38,33);(9,12,16,31)(10,18,22,39)(11,23,26,13)(14,30,15,29)(17,33,35,40)(19,37,36,34)(20,38,21,27)(24,32,25,28);(9,13)(10,19)(11,12)(14,32)(15,28)(16,23)(17,27)(18,37)(20,33)(21,40)(22,36)(24,29)(25,30)(26,31)(34,39)(35,38);(1,3,4,7)(2,6,5,8)(9,14,16,15)(10,20,22,21)(11,24,26,25)(12,29,31,30)(13,32,23,28)(17,34,35,37)(18,27,39,38)(19,33,36,40)

Polycyclic presentation:

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

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, 6346977);