#6484873

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470003
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1378 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)(3,6)(4,8)(5,7)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,23)(20,24)(25,26,31,32,30,36,29,35)(27,33,38,39,28,34,37,40);(1,3,5,4)(2,6,7,8)(9,11)(10,14)(12,19)(13,17)(15,23)(16,21)(18,20)(22,24)(26,32)(29,31)(33,39)(34,40)(35,36)(37,38);(1,4,5,3)(2,6,7,8)(11,17)(14,21)(18,19)(22,23)(25,27)(26,33)(28,30)(29,37)(31,38)(32,39)(34,36)(35,40);(9,12,13,20)(10,15,16,24)(11,18,17,19)(14,22,21,23)(25,28,30,27)(26,34,36,33)(29,38,31,37)(32,40,35,39);(10,16)(14,21)(15,24)(22,23)(27,28)(33,34)(37,38)(39,40)

Polycyclic presentation:

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

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, 6392069);