#354703

order	512
nilpotency class	3
isoclinism	isoclinic class of #3057
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×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	C₂⁴	C₂×#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1090 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)(9,10,16,20,14,19,15,31)(11,21,26,36,24,17,25,34)(12,27,29,35,28,40,13,18)(22,37,23,32,38,39,30,33);(2,4)(5,6,7,8)(9,11)(10,17)(12,22)(13,30)(14,24)(15,25)(16,26)(18,32)(19,21)(20,34)(23,29)(27,39)(28,38)(31,36)(33,35)(37,40);(9,12,15,13,14,28,16,29)(10,18,20,27,19,35,31,40)(11,22,25,30,24,38,26,23)(17,32,34,39,21,33,36,37);(9,13,15,28,14,29,16,12)(10,18,31,40,19,35,20,27)(11,23,25,22,24,30,26,38)(17,33,36,39,21,32,34,37)

Polycyclic presentation:

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

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, 261899);