#6373547

order	512
nilpotency class	3
isoclinism	isoclinic class of #60457
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

2036 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

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,e^d=ei

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6280743);