#556366

order	512
nilpotency class	2
isoclinism	isoclinic class of #37436
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

2166 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:

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

Polycyclic presentation:

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

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, 463562);