#6442675

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437537
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3500	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4644 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)(6,8)(9,10)(11,22)(12,18)(13,21)(14,20)(15,19)(16,24)(17,30)(23,38)(25,34)(26,33)(27,32)(28,36)(29,37)(31,39)(35,40);(2,4)(9,11,14,25)(10,16,20,33)(12,27,30,39)(13,26,15,24)(17,35,18,38)(19,34,21,22)(23,29,40,28)(31,37,32,36);(5,6,7,8)(9,12)(10,17)(11,23)(13,28)(14,30)(15,29)(16,31)(18,20)(19,36)(21,37)(22,35)(24,27)(25,40)(26,39)(32,33)(34,38);(9,12)(10,18)(11,23)(13,28)(14,30)(15,29)(16,32)(17,20)(19,37)(21,36)(22,38)(24,27)(25,40)(26,39)(31,33)(34,35);(9,13,14,15)(10,19,20,21)(11,24,25,26)(12,28,30,29)(16,22,33,34)(17,36,18,37)(23,27,40,39)(31,35,32,38)

Polycyclic presentation:

2^9|b^2=i,c^2=gi,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,h^a=hi,c^b=ch,d^b=dh,e^b=ei,f^b=fi,h^b=hi,f^c=fi,h^c=hi,f^d=fi,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, 6349871);