#498027

order	512
nilpotency class	3
isoclinism	isoclinic class of #189674
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#432
prime radical	C₂²×D₄	M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #1752 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

505 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=ef,e^2=i,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,f^a=fi,d^b=deh,g^b=gi,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405223);