#356581

order	512
nilpotency class	2
isoclinism	isoclinic class of #180873
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#376
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

875 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=ef,d^2=fg,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,h^a=hi,d^b=de,g^b=gi,h^d=hi

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 263777);