#218035

order	512
nilpotency class	3
isoclinism	isoclinic class of #32712
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³

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	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#431 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

528 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)(4,5)(7,8)(9,10)(11,16)(12,21)(13,18)(14,23)(15,19)(17,22)(20,24)(25,26,29,34,31,28,30,33)(27,32,36,39,37,38,35,40);(2,5)(3,7)(9,11,15,12)(10,16,19,21)(13,22,14,20)(17,23,24,18)(25,27,30,35,31,37,29,36)(26,32,33,40,28,38,34,39);(1,3,4,7)(2,6,5,8)(9,12,15,11)(10,17,19,24)(13,22,14,20)(16,23,21,18)(25,28,31,26)(27,32,37,38)(29,33,30,34)(35,40,36,39);(2,5)(6,8)(9,13,15,14)(10,18,19,23)(11,20,12,22)(16,24,21,17)(25,29,31,30)(26,33,28,34)(27,35,37,36)(32,39,38,40)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=eh,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dei,f^a=fh,c^b=cei,d^b=dh,d^c=di,f^c=fh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 125231);