#219369

order	512
nilpotency class	3
isoclinism	isoclinic class of #23918
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂²×D₈	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

690 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=ef,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce,d^c=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 126565);