#497088

order	512
nilpotency class	3
isoclinism	isoclinic class of #3186
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	C₂³	C₂×#131
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

442 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,10)(4,8)(5,14)(6,9)(7,12)(11,16)(13,15)(17,18,22,28,23,27,37,43,24,29,36,26,21,38,20,35)(19,25,32,41,33,40,46,48,34,42,45,39,31,47,30,44);(1,3)(2,7)(4,13)(5,11)(6,12)(8,16)(9,10)(14,15)(17,19,22,32,23,33,37,46,24,34,36,45,21,31,20,30)(18,25,28,41,27,40,43,48,29,42,26,39,38,47,35,44);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,10,16)(17,20,21,36,24,37,23,22)(18,26,27,35,29,28,38,43)(19,30,31,45,34,46,33,32)(25,39,40,44,42,41,47,48);(1,5,6,4)(2,8,9,14)(3,11,12,13)(7,16,10,15)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=ef,d^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

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, 404284);