#6454408

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437798
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#3006	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1867 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)(9,10,14,19,15,20,35,40)(11,21,25,37,26,36,24,16)(12,23,29,38,30,17,28,31)(13,18,32,27,33,22,34,39);(9,11,15,26)(10,16,20,37)(12,27,30,39)(13,31,33,38)(14,24,35,25)(17,34,23,32)(18,28,22,29)(19,36,40,21);(2,5)(6,8)(9,12)(10,17)(11,22)(13,16)(14,29)(15,30)(18,26)(19,31)(20,23)(21,32)(24,27)(25,39)(28,35)(33,37)(34,36)(38,40);(9,13)(10,18)(11,23)(12,16)(14,32)(15,33)(17,26)(19,27)(20,22)(21,29)(24,31)(25,38)(28,36)(30,37)(34,35)(39,40);(1,3,4,7)(2,6,5,8)(9,14,15,35)(10,19,20,40)(11,24,26,25)(12,28,30,29)(13,32,33,34)(16,36,37,21)(17,38,23,31)(18,27,22,39)

Polycyclic presentation:

2^9|a^2=f,b^2=i,e^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,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, 6361604);