#6560853

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535655
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1263 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=fgi,d^2=fhi,e^2=h,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=ei,e^d=eh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6468049);