#6380988

order	512
nilpotency class	3
isoclinism	isoclinic class of #61058
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×#575	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#432 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1906 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)(9,10,14,19,15,18,13,27)(11,20,23,16,21,32,28,30)(12,17,25,36,26,34,29,35)(22,37,39,31,38,33,24,40);(9,11)(10,16)(12,24)(13,28)(14,23)(15,21)(17,33)(18,30)(19,32)(20,27)(22,25)(26,39)(29,38)(31,35)(34,37)(36,40);(2,5)(6,8)(11,21)(16,30)(20,32)(22,38)(23,28)(24,39)(31,40)(33,37);(9,12)(10,17)(11,22)(13,29)(14,25)(15,26)(16,31)(18,34)(19,36)(20,37)(21,38)(23,39)(24,28)(27,35)(30,40)(32,33);(1,3,4,7)(2,6,5,8)(10,18)(16,30)(17,34)(19,27)(20,32)(31,40)(33,37)(35,36)

Polycyclic presentation:

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

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, 6288184);