#218781

order	512
nilpotency class	3
isoclinism	isoclinic class of #23924
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

611 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)(3,7)(4,6)(5,8)(10,15)(11,18)(12,14)(16,23)(19,24)(20,21)(25,26,29,33,31,34,30,38)(27,32,28,37,36,40,35,39);(1,3)(2,5)(4,8)(6,7)(9,10,13,15)(11,19,21,16)(12,22,14,17)(18,23,20,24)(25,27,30,35,31,36,29,28)(26,32,38,39,34,40,33,37);(3,8)(5,7)(9,11,12,20,13,21,14,18)(10,16,22,23,15,19,17,24)(25,28,29,36,31,35,30,27)(26,32,38,39,34,40,33,37);(2,6)(5,7)(9,12,13,14)(10,17,15,22)(11,20,21,18)(16,24,19,23)(25,29,31,30)(26,33,34,38)(27,35,36,28)(32,39,40,37)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 125977);