#369358

order	512
nilpotency class	3
isoclinism	isoclinic class of #196299
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1430 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)(3,7)(4,6)(5,8)(9,10)(11,16)(12,23,27,39)(13,18)(14,29)(15,19)(17,25,35,26)(20,36,37,34)(21,31)(22,33)(24,38,40,30)(28,32);(1,3,5,4)(2,6,8,7)(9,11,15,22)(10,16,19,33)(12,24)(13,28,14,21)(17,34)(18,32,29,31)(20,25)(23,38)(26,37)(27,40)(30,39)(35,36);(1,3,5,4)(2,7,8,6)(9,12)(10,17,19,35)(11,20,22,37)(13,25)(14,26)(15,27)(16,30)(18,23,29,39)(21,24,28,40)(31,36)(32,34)(33,38);(1,4,5,3)(2,7,8,6)

Polycyclic presentation:

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

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, 276554);