#6490553

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

1351 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)(4,5)(7,8)(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,23)(21,24)(25,26,30,36,31,37,29,35)(27,34,38,33,39,40,28,32);(9,11,14,21)(10,15,18,24)(12,20,13,19)(16,23,17,22)(25,27)(26,32)(28,30)(29,38)(31,39)(33,37)(34,35)(36,40);(2,5)(6,8)(9,12,14,13)(10,16,18,17)(11,19,21,20)(15,22,24,23)(25,28)(26,33)(27,30)(29,39)(31,38)(32,37)(34,35)(36,40);(9,13,14,12)(10,17,18,16)(11,20,21,19)(15,23,24,22)(25,27)(26,34)(28,29)(30,38)(31,39)(32,35)(33,36)(37,40);(1,3,4,7)(2,6,5,8)(10,18)(15,24)(16,17)(22,23)(25,29,31,30)(26,35,37,36)(27,38,39,28)(32,34,33,40)

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=h,d^2=h,e^2=g,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, 6397749);