#218463

order	512
nilpotency class	3
isoclinism	isoclinic class of #24378
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄

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×D₄	C₂²×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

618 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=h,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,f^c=fh

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, 125659);