#6497015

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470137
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 ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1370 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:

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

Polycyclic presentation:

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

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, 6404211);