#218803

order	512
nilpotency class	3
isoclinism	isoclinic class of #30396
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×C₄×M₁₆ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

630 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)(9,10)(11,18)(12,16)(13,22)(14,17)(15,20)(19,23)(21,24)(25,26,30,35,31,36,29,34)(27,32,37,40,38,33,28,39);(1,3)(2,5)(4,8)(6,7)(25,27,30,37,31,38,29,28)(26,32,35,40,36,33,34,39);(3,8)(5,7)(9,11,13,20,14,21,12,19)(10,15,16,18,17,23,22,24)(25,28,29,38,31,37,30,27)(26,33,35,39,36,32,34,40);(2,6)(5,7)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,18,23,24)(25,29,31,30)(26,34,36,35)(27,37,38,28)(32,40,33,39)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=f,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=di,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, 125999);