#481902

order	512
nilpotency class	4
isoclinism	isoclinic class of #481819
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

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

899 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=hi,d^2=i,e^2=h,b^a=be,c^a=ch,e^a=eg,g^a=gi,d^b=di,e^b=eh,f^b=fgh,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, 389098);