#6478162

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470185
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 ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1347 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)(3,8)(4,6)(5,7)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,23)(20,24)(25,26,32,38)(27,33,39,34)(28,40,29,35)(30,37,31,36);(1,3,5,4)(2,6,7,8)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,27)(26,33)(28,31)(29,30)(32,39)(34,38)(35,37)(36,40);(9,12,13,20)(10,15,16,24)(11,17,18,19)(14,21,22,23)(25,28,32,29)(26,34,38,33)(27,30,39,31)(35,36,40,37);(1,4,5,3)(2,6,7,8)(11,18)(14,22)(17,19)(21,23)(25,29,32,28)(26,35,38,40)(27,30,39,31)(33,37,34,36);(10,16)(14,22)(15,24)(21,23)(25,30,32,31)(26,36,38,37)(27,28,39,29)(33,35,34,40)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=h,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dhi,e^b=ei,g^c=gi,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, 6385358);