#353378

order	512
nilpotency class	3
isoclinism	isoclinic class of #3008
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×M₁₆ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

4066 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,d^b=di,e^b=ei,f^b=fi,e^c=ei,f^c=fi

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 260574);