#482682

order	512
nilpotency class	4
isoclinism	isoclinic class of #482397
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×C₂²⋊C₄	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#520 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

887 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)(3,4)(7,8)(9,10)(11,19)(12,23,24,16)(13,26,27,17)(14,18)(15,21,29,36)(20,33,34,31)(22,32)(25,30)(28,35)(37,40)(38,39);(1,2)(3,6)(4,8)(5,7)(9,11,14,22)(10,15,18,29)(12,20,24,34)(13,21,27,36)(16,28,23,37)(17,19,26,32)(25,35,39,40)(30,31,38,33);(1,3,5,4)(2,6,7,8)(12,24)(16,23)(20,34)(25,39)(28,37)(30,38)(31,33)(35,40);(1,3,5,4)(2,6,7,8)(9,12,14,24)(10,16,18,23)(11,20,22,34)(13,25,27,39)(15,28,29,37)(17,30,26,38)(19,31,32,33)(21,35,36,40)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=h,d^2=hi,e^2=h,b^a=be,c^a=ch,d^a=dg,e^a=eg,g^a=gi,e^b=eh,f^b=fgh,d^c=di,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, 389878);