#562568

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

2114 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)(9,10,13,14)(11,17,18,15)(12,20,21,16)(19,22,24,23)(25,26,31,29)(27,32,28,35)(30,37,38,33)(34,40,36,39);(6,8)(10,14)(11,18)(12,21)(22,23)(25,27,31,28)(26,32,29,35)(30,34,38,36)(33,39,37,40);(5,6)(7,8)(9,10)(11,17)(12,16)(13,14)(15,18)(19,23)(20,21)(22,24)(25,28,31,27)(26,32,29,35)(30,36,38,34)(33,39,37,40);(1,3,4,2)(9,11,13,18)(10,15,14,17)(12,19,21,24)(16,22,20,23)(25,29,31,26)(27,32,28,35)(30,37,38,33)(34,39,36,40);(9,12,13,21)(10,16,14,20)(11,19,18,24)(15,22,17,23)(25,30)(26,33)(27,34)(28,36)(29,37)(31,38)(32,39)(35,40)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=fg,d^2=i,e^2=f,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, 469764);