#6438688

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×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3178 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=fi,c^2=f,d^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,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, 6345884);