#7514051

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

3163 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)(9,10,14,19)(11,20,21,16)(12,24,26,33)(13,15,27,18)(17,28,32,25)(22,37,35,40)(23,31,36,29)(30,38,34,39);(1,2)(3,4)(5,6)(7,8)(10,15)(13,27)(16,29)(17,24)(18,19)(20,31)(23,36)(25,28)(30,40)(32,33)(34,37)(38,39);(6,8)(9,11)(10,16)(12,22)(13,23)(14,21)(15,29)(17,30)(18,31)(19,20)(24,40)(25,38)(26,35)(27,36)(28,39)(32,34)(33,37);(2,4)(6,8)(11,21)(16,20)(22,35)(23,36)(29,31)(30,34)(37,40)(38,39);(9,12,14,26)(10,17,19,32)(11,22,21,35)(13,28,27,25)(15,24,18,33)(16,30,20,34)(23,39,36,38)(29,40,31,37)

Polycyclic presentation:

2^9|a^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,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, 7421247);