#219251

order	512
nilpotency class	3
isoclinism	isoclinic class of #30396
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 ◁ #1744 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

565 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)(11,17)(12,21)(13,16)(14,23)(15,19)(18,24)(20,22)(25,26,30,35,29,33,36,34)(27,32,38,40,37,31,28,39);(1,2)(3,6)(4,5)(7,8)(9,11,13,19)(10,15,16,17)(12,22,23,18)(14,20,21,24)(25,27,36,28,29,37,30,38)(26,31,34,40,33,32,35,39);(2,5)(6,8)(9,12,21,10,13,23,14,16)(11,18,24,17,19,22,20,15)(25,28,36,37,29,38,30,27)(26,32,35,40,33,31,34,39);(1,3,4,7)(2,6,5,8)(10,16)(11,19)(12,23)(20,24)(26,33)(31,32)(34,35)(39,40)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=fh,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^b=dhi,d^c=dhi

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, 126447);