#6373534

order	512
nilpotency class	3
isoclinism	isoclinic class of #60460
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×C₄×D₄
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2030 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=f,d^2=h,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ci,e^b=ei,e^c=ei,e^d=ei

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, 6280730);