#482376

order	512
nilpotency class	4
isoclinism	isoclinic class of #482362
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1017 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)(3,5)(4,6)(7,8)(9,10)(11,21,25,32)(12,26)(13,19,14,18)(15,20)(16,24)(17,30)(22,38,27,37)(23,33)(28,35,29,36)(31,39)(34,40);(2,5)(3,8)(6,7)(9,11,15,25)(10,16,20,33)(12,27,30,22)(13,24,14,23)(17,34,26,31)(18,21,19,32)(28,39,29,40)(35,38,36,37);(2,6)(5,7)(9,12,15,30)(10,17,20,26)(11,22,25,27)(13,28,14,29)(16,31,33,34)(18,35,19,36)(21,37,32,38)(23,39,24,40);(2,6)(5,7)(9,13,15,14)(10,18,20,19)(11,23,25,24)(12,28,30,29)(16,32,33,21)(17,35,26,36)(22,39,27,40)(31,38,34,37)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=i,e^2=h,b^a=be,c^a=ch,d^a=dg,e^a=eg,g^a=gi,c^b=ci,d^b=di,e^b=eh,f^b=fgh,f^d=fi,f^e=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 389572);