#6447745

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2967 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fg,c^2=g,d^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, 6354941);