#6410373

order	512
nilpotency class	2
isoclinism	isoclinic class of #45438
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1444 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=fg,d^2=f,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=efi,c^b=cgi,d^b=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, 6317569);