#481806

order	512
nilpotency class	4
isoclinism	isoclinic class of #481619
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄⋊C₄	C₂⁴
prime radical	#2619	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#519 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

911 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 389002);