#354627

order	512
nilpotency class	3
isoclinism	isoclinic class of #468
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₈	C₂×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1618 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)(6,8)(9,10,14,18,15,19,13,28)(11,20,24,29,21,33,23,16)(12,17,26,31,27,32,25,38)(22,34,37,39,35,40,36,30);(2,4)(9,11)(10,16,19,29)(12,22)(13,23)(14,24)(15,21)(17,30,32,39)(18,20,28,33)(25,36)(26,37)(27,35)(31,34,38,40);(5,6,7,8)(11,21)(16,29)(20,33)(22,35)(23,24)(30,39)(34,40)(36,37);(9,12,14,26,15,27,13,25)(10,17,18,31,19,32,28,38)(11,22,24,37,21,35,23,36)(16,30,20,34,29,39,33,40)

Polycyclic presentation:

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

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, 261823);