#496856

order	512
nilpotency class	3
isoclinism	isoclinic class of #3198
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	#525	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×M₃₂ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

515 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)(4,5)(7,8)(9,10,15,21,16,20,33,38,13,19,29,26,14,30,32,39)(11,12,24,28,25,27,18,36,22,17,34,35,23,31,37,40);(2,5)(3,7)(9,11,15,24,16,25,33,18,13,22,29,34,14,23,32,37)(10,17,21,35,20,31,38,40,19,12,26,28,30,27,39,36);(1,3,4,7)(2,6,5,8)(9,12,13,17)(10,18,19,37)(11,21,22,26)(14,31,16,27)(15,28,29,35)(20,34,30,24)(23,39,25,38)(32,40,33,36);(2,5)(6,8)(10,19)(11,22)(18,37)(20,30)(21,26)(23,25)(24,34)(38,39)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404052);