#6442866

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2586 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6350062);