#7514116

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

3177 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)(11,21)(12,18)(13,19)(14,29)(15,30)(16,20)(17,23)(22,36)(24,33)(25,39)(26,35)(27,34)(28,32)(31,38)(37,40);(1,2)(3,4)(5,6)(7,8)(9,11,16,14)(10,17,20,32)(12,22,26,25)(13,23,15,28)(18,31,35,40)(19,29,30,21)(24,37,27,38)(33,36,34,39);(6,8)(9,12)(10,18)(11,22)(13,27)(14,25)(15,24)(16,26)(17,31)(19,34)(20,35)(21,36)(23,38)(28,37)(29,39)(30,33)(32,40);(2,4)(6,8)(9,13,16,15)(10,19,20,30)(11,23,14,28)(12,24,26,27)(17,29,32,21)(18,33,35,34)(22,37,25,38)(31,36,40,39);(9,14,16,11)(10,17,20,32)(12,25,26,22)(13,28,15,23)(18,31,35,40)(19,29,30,21)(24,38,27,37)(33,36,34,39)

Polycyclic presentation:

2^9|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=cg,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, 7421312);