#356210

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

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	C₂²×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄⋊C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1444 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)(5,6,8,7)(9,10,13,20,15,18,14,19)(11,16,22,32,24,30,23,31)(12,25,27,33,28,34,26,17)(21,35,37,39,38,40,36,29);(6,7)(9,11,14,23,15,24,13,22)(10,16,19,31,18,30,20,32)(12,21,26,36,28,38,27,37)(17,29,34,40,33,39,25,35);(9,12)(10,17)(11,21)(13,27)(14,26)(15,28)(16,29)(18,33)(19,34)(20,25)(22,37)(23,36)(24,38)(30,39)(31,40)(32,35);(5,7,8,6)(10,18)(16,30)(17,33)(19,20)(25,34)(29,39)(31,32)(35,40)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 263406);