#497984

order	512
nilpotency class	4
isoclinism	isoclinic class of #484095
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#442
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1770 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

531 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=fi,d^2=i,e^2=h,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^b=de,e^b=ehi,g^b=gi,h^b=hi,e^d=ehi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405180);