#6445218

order	512
nilpotency class	3
isoclinism	isoclinic class of #6439029
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

1964 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)(3,6)(4,5)(7,8)(9,10)(11,21)(12,25)(13,17)(14,18)(15,20)(16,23)(19,31)(22,38)(24,36)(26,33)(27,40)(28,34)(29,37)(30,35)(32,39);(2,5)(6,8)(9,11)(10,16)(12,15)(13,27)(14,23)(17,33)(18,21)(19,25)(20,36)(22,29)(24,31)(26,28)(30,39)(32,37)(34,40)(35,38);(9,12,15,23,14,24,31,11)(10,16,19,25,18,21,20,36)(13,26,29,39,28,27,30,22)(17,32,35,33,34,38,37,40);(9,13)(10,17)(11,22)(12,26)(14,28)(15,29)(16,32)(18,34)(19,35)(20,37)(21,38)(23,39)(24,27)(25,33)(30,31)(36,40);(1,3,4,7)(2,6,5,8)(11,23)(12,24)(16,21)(22,39)(25,36)(26,27)(32,38)(33,40)

Polycyclic presentation:

2^9|c^2=g,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dhi,e^b=ei,h^b=hi,e^c=ei,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, 6352414);