#557082

order	512
nilpotency class	2
isoclinism	isoclinic class of #45438
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)(9,10)(11,17)(12,16)(13,14)(15,18)(19,23)(20,21)(22,24)(26,30)(27,33)(28,35)(38,40);(2,4)(10,14)(11,18)(12,20)(22,23)(25,26,29,30)(27,31,33,37)(28,36,35,32)(34,40,39,38);(1,2,3,4)(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24)(25,26)(27,31)(28,32)(29,30)(33,37)(34,38)(35,36)(39,40);(9,11,13,18)(10,15,14,17)(12,19,20,24)(16,22,21,23)(25,27)(26,31)(28,34)(29,33)(30,37)(32,38)(35,39)(36,40);(9,12,13,20)(10,16,14,21)(11,19,18,24)(15,22,17,23)(25,28,29,35)(26,32,30,36)(27,34,33,39)(31,38,37,40)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=fh,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dfg,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, 464278);