#6499224

order	512
nilpotency class	3
isoclinism	isoclinic class of #6469956
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×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1384 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=gh,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,d^c=di

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, 6406420);