#7514311

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

3160 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)(9,10,15,20)(11,21,24,17)(12,19,14,16)(13,27,28,31)(18,26,34,29)(22,32,25,30)(23,37,39,40)(33,36,35,38);(1,2)(3,4)(5,6)(7,8)(10,16)(12,14)(17,30)(18,31)(19,20)(21,32)(22,25)(26,29)(27,34)(33,37)(35,40)(36,38);(6,8)(9,11)(10,17)(12,25)(13,23)(14,22)(15,24)(16,30)(18,33)(19,32)(20,21)(26,38)(27,40)(28,39)(29,36)(31,37)(34,35);(2,4)(6,8)(9,12,15,14)(10,16,20,19)(11,22,24,25)(13,26,28,29)(17,32,21,30)(18,31,34,27)(23,36,39,38)(33,40,35,37);(9,13,15,28)(10,18,20,34)(11,23,24,39)(12,26,14,29)(16,31,19,27)(17,33,21,35)(22,36,25,38)(30,37,32,40)

Polycyclic presentation:

2^9|a^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, 7421507);