#212740

order	512
nilpotency class	3
isoclinism	isoclinic class of #23387
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

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

594 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)(12,19)(15,22)(16,21)(18,23)(25,26,30,34,31,36,29,35)(27,37,28,33,39,40,38,32);(2,5)(6,8)(9,10,12,14,13,16,19,21)(11,15,17,20,18,22,23,24)(25,27,30,28,31,39,29,38)(26,32,35,40,36,33,34,37);(1,3,4,7)(2,6,5,8)(9,11,12,17,13,18,19,23)(10,15,14,20,16,22,21,24)(25,27,29,38,31,39,30,28)(26,33,34,40,36,32,35,37);(10,16)(14,21)(15,22)(20,24)(25,28,30,39,31,38,29,27)(26,33,34,40,36,32,35,37)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=e,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, 119936);