#244435

order	512
nilpotency class	3
isoclinism	isoclinic class of #232749
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2397
prime radical	#3150	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

605 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=h,c^2=h,d^2=ef,e^2=h,f^2=i,b^a=be,c^a=cf,g^a=ghi,c^b=cg,e^b=eh,f^b=fh,d^c=dh,e^c=ei,f^c=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151631);