#212683

order	512
nilpotency class	3
isoclinism	isoclinic class of #24378
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, prime radical	C₂³	C₂×C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

576 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 119879);