#123746

order	512
nilpotency class	3
isoclinism	isoclinic class of #2031
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#124
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂³	#397
prime radical	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

1474 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=d,d^2=i,e^2=i,g^2=i,b^a=bd,c^a=ce,e^a=ei,h^a=hi,c^b=cf,f^b=fi,g^b=gi,e^c=ei,h^c=hi

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, 30942);