#391607

order	512
nilpotency class	3
isoclinism	isoclinic class of #232181
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2380
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

686 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=e,d^2=eg,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,g^a=gi,c^b=cg,d^b=def,e^b=ei,f^b=fi,g^b=gi,d^c=de,h^c=hi,e^d=ei,f^d=fi,g^d=gi

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, 298803);