#544498

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
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

2190 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)(9,10,14,11)(12,15,20,17)(13,18,21,16)(19,23,24,22)(26,27)(29,37)(31,33)(35,40);(2,4)(10,11)(15,17)(16,18)(22,23)(25,26,30,27)(28,33,36,31)(29,32,37,34)(35,39,40,38);(1,2)(3,4)(9,11,14,10)(12,15,20,17)(13,18,21,16)(19,22,24,23)(25,27,30,26)(28,31,36,33)(29,34,37,32)(35,38,40,39);(9,12)(10,15)(11,17)(13,19)(14,20)(16,22)(18,23)(21,24)(25,28,30,36)(26,31,27,33)(29,35,37,40)(32,38,34,39);(9,13,14,21)(10,16,11,18)(12,19,20,24)(15,22,17,23)(25,29)(26,32)(27,34)(28,35)(30,37)(31,38)(33,39)(36,40)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=f,d^2=g,e^2=fg,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dg,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, 451694);