#566851

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

2170 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)(9,10)(11,16)(12,18)(13,17)(14,15)(19,23)(20,24)(21,22)(26,30)(27,33)(28,36)(38,40);(6,8)(10,15)(11,18)(13,21)(20,23)(25,26,29,30)(27,34,33,31)(28,37,36,32)(35,38,39,40);(1,2,4,3)(5,6)(7,8)(9,11,14,18)(10,16,15,12)(13,19,21,24)(17,23,22,20)(25,26)(27,34)(28,32)(29,30)(31,33)(35,40)(36,37)(38,39);(1,3,4,2)(9,12,14,16)(10,11,15,18)(13,20,21,23)(17,19,22,24)(25,27)(26,31)(28,35)(29,33)(30,34)(32,38)(36,39)(37,40);(9,13)(10,17)(11,19)(12,20)(14,21)(15,22)(16,23)(18,24)(25,28,29,36)(26,32,30,37)(27,35,33,39)(31,38,34,40)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=fgi,d^2=fgi,e^2=g,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, 474047);