#107887

order	512
nilpotency class	3
isoclinism	isoclinic class of #8479
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#379
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₂²⋊#95
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

710 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=i,e^2=i,f^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,d^b=di,e^b=eh,f^b=fhi,d^c=dh,e^c=ei,f^c=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 15083);