#6569254

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535815
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

1262 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=gi,c^2=f,d^2=fgi,e^2=fgi,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=ei,d^c=di,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, 6476450);