#6438724

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×#3010	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3198 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)(13,18)(14,20)(15,19)(21,31)(22,33)(23,32)(24,34)(25,36)(26,35)(27,37)(28,39)(29,40)(30,38);(5,6,8,7)(9,11,15,22)(10,16,19,33)(12,24,25,27)(13,26,21,29)(14,28,30,23)(17,34,36,37)(18,35,31,40)(20,39,38,32);(2,4)(5,7,8,6)(9,12)(10,17)(11,21)(13,22)(14,29)(15,25)(16,31)(18,33)(19,36)(20,40)(23,27)(24,28)(26,30)(32,37)(34,39)(35,38);(9,13,15,21)(10,18,19,31)(11,12,22,25)(14,24,30,27)(16,17,33,36)(20,34,38,37)(23,29,28,26)(32,40,39,35);(10,19)(11,22)(12,25)(18,31)(20,38)(23,28)(26,29)(34,37)

Polycyclic presentation:

2^9|b^2=fi,c^2=f,d^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,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, 6345920);