#6554954

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
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	C₂×#127	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1389 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)(10,14)(11,17)(16,21)(19,23)(25,26,29,33,30,34,40,32)(27,35,36,38,31,28,37,39);(1,3,5,4)(2,6,8,7)(9,10,13,14)(11,18,17,15)(12,16,20,21)(19,24,23,22)(25,27,30,31)(26,28,34,35)(29,36,40,37)(32,38,33,39);(1,3,5,4)(2,7,8,6)(9,11,13,17)(10,15,14,18)(12,19,20,23)(16,22,21,24)(25,28)(26,31)(27,34)(29,38)(30,35)(32,37)(33,36)(39,40);(1,4,5,3)(2,6,8,7)(12,20)(16,21)(19,23)(22,24)(25,29,30,40)(26,32,34,33)(27,36,31,37)(28,38,35,39);(9,12,13,20)(10,16,14,21)(11,19,17,23)(15,22,18,24)(25,29,30,40)(26,33,34,32)(27,37,31,36)(28,38,35,39)

Polycyclic presentation:

2^9|a^2=fgi,b^2=f,c^2=fi,d^2=fh,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ei,g^c=gi,e^d=eh

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6462150);