#6445207

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	#24292	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1942 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)(11,16)(12,24)(13,28)(14,18)(15,19)(17,26)(20,34)(21,23)(22,32)(25,40)(27,35)(29,33)(30,36)(31,39)(37,38);(2,5)(6,8)(9,11)(10,16)(12,25)(13,23)(14,29)(15,22)(17,34)(18,33)(19,32)(20,26)(21,28)(24,40)(27,39)(30,38)(31,35)(36,37);(9,12,13,26,15,27,14,30)(10,17,18,24,19,36,28,35)(11,20,23,39,22,38,29,25)(16,31,33,34,32,40,21,37);(9,10)(11,21)(12,17)(13,18)(14,28)(15,19)(16,23)(20,37)(22,33)(24,26)(25,40)(27,36)(29,32)(30,35)(31,39)(34,38);(1,3,4,7)(2,6,5,8)(11,22)(16,32)(20,38)(21,33)(23,29)(25,39)(31,40)(34,37)

Polycyclic presentation:

2^9|c^2=g,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,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, 6352403);