#212741

order	512
nilpotency class	3
isoclinism	isoclinic class of #23387
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×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	C₂×#127	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ #1744 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

650 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)(25,26,29,33,30,34,36,32)(27,35,28,31,37,39,38,40);(2,5)(6,8)(9,10,12,14,13,16,18,21)(11,15,17,20,19,22,23,24)(25,27,29,28,30,37,36,38)(26,31,32,35,34,40,33,39);(1,3,4,7)(2,6,5,8)(9,11,18,23,13,19,12,17)(10,15,21,24,16,22,14,20)(25,27,36,38,30,37,29,28)(26,31,33,39,34,40,32,35);(10,16)(14,21)(15,22)(20,24)(25,28,29,37,30,38,36,27)(26,31,33,39,34,40,32,35)

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,e^a=eh,f^a=fh,c^b=ci,d^b=dh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 119937);