#513866

order	512
nilpotency class	2
isoclinism	isoclinic class of #3532
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
commutator	C₂²	C₂³×C₄²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2518 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=fgh,d^2=fh,e^2=fg,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=df

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 421062);