#7514298

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

3189 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:

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

Polycyclic presentation:

2^9|c^2=i,d^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, 7421494);