#6449274

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437969
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#60644	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3355 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)(3,6)(4,5)(7,8)(9,10)(11,21)(12,17)(13,18)(14,19)(15,30)(16,20)(22,37)(23,29)(24,31)(25,33)(26,34)(27,36)(28,35)(32,38)(39,40);(9,11,15,18,16,24,14,23)(10,13,19,21,20,29,30,31)(12,22,27,38,28,39,26,33)(17,32,34,37,35,25,36,40);(2,5)(6,8)(9,12,16,28)(10,17,20,35)(11,22,24,39)(13,25,29,32)(14,27,15,26)(18,33,23,38)(19,36,30,34)(21,37,31,40);(1,3,4,7)(2,6,5,8)(9,13)(10,18)(11,19)(12,25)(14,21)(15,31)(16,29)(17,33)(20,23)(22,36)(24,30)(26,40)(27,37)(28,32)(34,39)(35,38);(9,14,16,15)(10,19,20,30)(11,23,24,18)(12,26,28,27)(13,21,29,31)(17,34,35,36)(22,33,39,38)(25,40,32,37)

Polycyclic presentation:

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

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, 6356470);