#6445219

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
prime radical	#24294	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1969 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)(11,17)(12,24)(13,19)(14,29)(15,20)(16,21)(18,23)(22,31)(25,34)(26,32)(27,36)(28,37)(30,35)(33,38)(39,40);(2,5)(6,8)(9,11)(10,17)(12,14)(13,25)(15,26)(16,23)(18,21)(19,34)(20,32)(22,30)(24,29)(27,38)(28,39)(31,35)(33,36)(37,40);(9,12,15,11,16,26,14,23)(10,18,29,32,21,17,20,24)(13,25,30,22,28,39,27,38)(19,33,36,40,37,31,35,34);(9,13)(10,19)(11,22)(12,25)(14,27)(15,30)(16,28)(17,31)(18,33)(20,35)(21,37)(23,38)(24,34)(26,39)(29,36)(32,40);(1,3,4,7)(2,6,5,8)(11,23)(12,26)(17,18)(22,38)(24,32)(25,39)(31,33)(34,40)

Polycyclic presentation:

2^9|c^2=gi,e^2=f,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, 6352415);