#6544625

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
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	#1999	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1386 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)(11,17,18,14)(12,15,20,22)(19,23,24,21)(26,30)(28,36)(31,38)(32,37)(33,39)(35,40);(2,5)(6,8)(9,11)(10,14)(12,19)(13,18)(15,21)(16,17)(20,24)(22,23)(25,26,29,32)(27,33,34,31)(28,37,36,30)(35,38,40,39);(11,18)(14,17)(19,24)(21,23)(25,26)(27,31)(28,37)(29,32)(30,36)(33,34)(35,39)(38,40);(1,3,4,7)(2,6,5,8)(12,20)(15,22)(19,24)(21,23)(25,27)(26,31)(28,35)(29,34)(30,38)(32,33)(36,40)(37,39);(9,12,13,20)(10,15,16,22)(11,19,18,24)(14,21,17,23)(27,34)(31,33)(35,40)(38,39)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6451821);