#6445191

order	512
nilpotency class	3
isoclinism	isoclinic class of #6439025
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	#24291	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1967 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)(11,15)(12,25)(13,18)(14,19)(16,28)(17,27)(20,37)(21,23)(22,32)(24,33)(26,31)(29,35)(30,36)(34,38)(39,40);(2,5)(6,8)(9,11)(10,15)(12,26)(13,23)(14,22)(16,34)(17,33)(18,21)(19,32)(20,29)(24,27)(25,31)(28,38)(30,39)(35,37)(36,40);(9,12,27,29,14,30,13,28)(10,16,18,36,19,35,17,25)(11,20,24,39,22,38,23,26)(15,31,21,34,32,40,33,37);(9,10)(11,21)(12,16)(13,17)(14,19)(15,23)(18,27)(20,34)(22,33)(24,32)(25,28)(26,31)(29,36)(30,35)(37,38)(39,40);(1,3,4,7)(2,6,5,8)(11,22)(15,32)(20,38)(21,33)(23,24)(26,39)(31,40)(34,37)

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=dh,e^b=ei,h^b=hi,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, 6352387);