#495973

order	512
nilpotency class	3
isoclinism	isoclinic class of #3008
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₈×D₄
prime radical	C₂×#41	C₂²×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#346 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

532 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)(3,8)(4,6)(5,7)(9,10,15,21,16,17,33,34,14,20,30,37,11,22,25,39)(12,26,28,24,27,18,35,13,19,32,38,29,23,36,40,31);(1,3,5,4)(2,6,7,8)(9,11,14,16)(10,17,20,22)(12,23,19,27)(13,29,31,24)(15,25,30,33)(18,32,36,26)(21,34,37,39)(28,40,38,35);(9,12)(10,18)(11,23)(13,21)(14,19)(15,28)(16,27)(17,32)(20,36)(22,26)(24,39)(25,40)(29,34)(30,38)(31,37)(33,35);(1,4,5,3)(2,6,7,8)(9,13,30,26,16,29,25,18,14,31,15,32,11,24,33,36)(10,19,37,28,17,23,39,35,20,12,21,38,22,27,34,40)

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403169);