#6452233

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	#60456	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3206 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)(9,10,15,21)(11,18,25,37)(12,17,29,36)(13,19,14,20)(16,23,31,24)(22,33,38,32)(26,30,39,40)(27,34,28,35);(9,11,14,24,15,25,13,23)(10,16,19,18,21,31,20,37)(12,26,27,38,29,39,28,22)(17,32,35,30,36,33,34,40);(1,3,5,4)(2,6,7,8)(9,12)(10,17)(11,22)(13,28)(14,27)(15,29)(16,30)(18,33)(19,35)(20,34)(21,36)(23,39)(24,26)(25,38)(31,40)(32,37);(1,4,5,3)(2,6,7,8)(9,11)(10,18)(12,22)(13,23)(14,24)(15,25)(16,20)(17,33)(19,31)(21,37)(26,27)(28,39)(29,38)(30,34)(32,36)(35,40);(9,13,15,14)(10,19,21,20)(11,23,25,24)(12,27,29,28)(16,18,31,37)(17,34,36,35)(22,26,38,39)(30,32,40,33)

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,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, 6359429);