#6452237

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437935
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#60454	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3246 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)(3,8)(4,6)(5,7)(9,10,16,22)(11,23,12,18)(13,19,31,36)(14,20,15,21)(17,26,33,25)(24,38,27,34)(28,32,39,40)(29,35,30,37);(9,11,15,25,16,12,14,26)(10,17,20,23,22,33,21,18)(13,28,30,27,31,39,29,24)(19,34,35,32,36,38,37,40);(1,3,5,4)(2,6,7,8)(9,12)(10,18)(11,16)(13,27)(14,25)(15,26)(17,20)(19,34)(21,33)(22,23)(24,31)(28,29)(30,39)(32,35)(36,38)(37,40);(1,4,5,3)(2,6,7,8)(9,13)(10,19)(11,24)(12,27)(14,29)(15,30)(16,31)(17,32)(18,34)(20,35)(21,37)(22,36)(23,38)(25,28)(26,39)(33,40);(9,14,16,15)(10,20,22,21)(11,25,12,26)(13,29,31,30)(17,18,33,23)(19,35,36,37)(24,28,27,39)(32,34,40,38)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=g,d^2=g,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,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.

Relationships

maximal subgroup of	#72286186, #72286194, #72286224
maximal normal subgroup of	#72286186, #72286194, #72286224

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6359433);