#354706

order	512
nilpotency class	3
isoclinism	isoclinic class of #3055
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×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1098 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)(12,27,30,35,29,39,13,18)(22,37,23,32,28,40,38,33);(2,4)(5,6,7,8)(9,11)(10,17)(12,28)(13,23)(14,24)(15,25)(16,26)(18,33)(19,21)(20,34)(22,29)(27,37)(30,38)(31,36)(32,35)(39,40);(9,12,15,13,14,29,16,30)(10,18,20,27,19,35,31,39)(11,22,25,38,24,28,26,23)(17,32,34,40,21,33,36,37);(9,13,15,29,14,30,16,12)(10,18,31,39,19,35,20,27)(11,23,25,22,24,38,26,28)(17,33,36,40,21,32,34,37)

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,c^b=ci,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, 261902);