#212744

order	512
nilpotency class	3
isoclinism	isoclinic class of #29736
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂×C₄²

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	C₂×#127	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ #1744 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

608 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)(10,14)(11,17)(12,18)(15,22)(16,21)(19,23)(25,26,29,34,30,35,37,33)(27,36,28,31,38,40,39,32);(2,5)(6,8)(9,10,12,14,13,16,18,21)(11,15,17,20,19,22,23,24)(25,27,29,28,30,38,37,39)(26,31,33,36,35,32,34,40);(1,3,4,7)(2,6,5,8)(9,11,18,23,13,19,12,17)(10,15,21,24,16,22,14,20)(25,27,37,39,30,38,29,28)(26,31,34,40,35,32,33,36);(10,16)(14,21)(15,22)(20,24)(25,28,29,38,30,39,37,27)(26,32,34,36,35,31,33,40)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 119940);