#482696

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×C₄⋊C₄	C₂⁴
prime radical	#392	C₂×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

866 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)(9,10,13,18,14,15,27,29)(11,16,20,31,21,28,37,38)(12,22,24,34,25,17,26,33)(19,32,35,40,23,30,36,39);(1,2)(3,6)(4,8)(5,7)(10,15)(16,28)(17,22)(30,32);(1,3,5,4)(2,6,7,8)(9,11,14,21)(10,16,15,28)(12,23,25,19)(13,20,27,37)(17,32,22,30)(18,31,29,38)(24,36,26,35)(33,40,34,39);(1,3,5,4)(2,6,7,8)(9,12,14,25)(10,17,15,22)(11,19,21,23)(13,26,27,24)(16,30,28,32)(18,34,29,33)(20,36,37,35)(31,40,38,39)

Polycyclic presentation:

2^9|a^2=f,c^2=hi,d^2=hi,e^2=h,f^2=i,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, 389892);