#481810

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	C₂⁴	C₂×C₂²⋊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

871 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,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, 389006);