#554917

order	512
nilpotency class	2
isoclinism	isoclinic class of #36702
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2122 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)(10,14)(15,20)(16,21)(22,24)(26,30)(27,33)(32,36)(35,39);(6,8)(10,14)(11,17)(16,21)(18,23)(25,26,29,30)(27,34,33,31)(28,36,37,32)(35,38,39,40);(5,6)(7,8)(9,10,13,14)(11,15,17,20)(12,16,19,21)(18,22,23,24)(25,26)(27,31)(28,32)(29,30)(33,34)(35,38)(36,37)(39,40);(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24)(25,27,29,33)(26,31,30,34)(28,35,37,39)(32,38,36,40);(1,3,4,2)(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(25,28,29,37)(26,32,30,36)(27,35,33,39)(31,38,34,40)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fg,d^2=f,e^2=gi,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=dg,e^b=ef

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 462113);