#243853

order	512
nilpotency class	3
isoclinism	isoclinic class of #232755
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	C₂×#399
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

641 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=ef,c^2=h,d^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dh,g^a=ghi,c^b=cg,d^b=dh,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, 151049);