#6448077

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438880
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1946 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=g,c^2=fg,d^2=f,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,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, 6355273);