#7514319

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3139 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,f^a=fi,c^b=cgi,d^b=dh,f^b=fi,d^c=di,f^e=fi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7421515);