#497746

order	512
nilpotency class	3
isoclinism	isoclinic class of #188770
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	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #1752 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

465 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)(9,10,14,20,13,18,27,17,15,19,30,35,28,36,12,26)(11,21,25,39,24,33,22,37,23,38,29,32,40,16,31,34);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16,19,33)(12,22)(13,24)(14,29)(15,23)(17,32,26,39)(18,21,36,38)(20,37,35,34)(25,30)(27,31)(28,40);(1,3,4,7)(2,6,5,8)(9,12,13,14,15,27,28,30)(10,17,18,35,19,26,36,20)(11,22,24,29,23,31,40,25)(16,32,21,34,33,39,38,37);(2,5)(6,8)(11,23)(16,33)(21,38)(22,31)(24,40)(25,29)(32,39)(34,37)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404942);