#563222

order	512
nilpotency class	2
isoclinism	isoclinic class of #47826
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

2094 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)(12,19)(15,20)(18,23)(25,26,31,28)(27,33,29,32)(30,34,38,36)(35,40,37,39);(6,8)(10,14)(11,17)(16,21)(18,23)(25,27,31,29)(26,32,28,33)(30,37,38,35)(34,40,36,39);(5,6,7,8)(9,10,13,14)(11,15,17,20)(12,16,19,21)(18,22,23,24)(25,28,31,26)(27,32,29,33)(30,36,38,34)(35,39,37,40);(1,3,4,2)(9,11,13,17)(10,15,14,20)(12,18,19,23)(16,22,21,24)(25,29,31,27)(26,33,28,32)(30,37,38,35)(34,40,36,39);(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(25,30,31,38)(26,34,28,36)(27,35,29,37)(32,39,33,40)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=h,d^2=gi,e^2=fg,b^a=bf,c^a=cg,d^a=df,e^a=eg,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, 470418);