#243905

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	C₂×#399
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

713 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)(3,11,12,16,14,15,13,7)(17,18,21,26,22,27,23,38)(19,28,31,41,32,40,30,24)(20,33,36,43,37,25,35,44)(29,45,34,48,46,42,47,39);(1,3)(2,7)(4,13)(5,12)(6,14)(8,15)(9,11)(10,16)(17,19,23,31,22,32,21,30)(18,24,38,28,27,41,26,40)(20,34,35,46,37,47,36,29)(25,42,43,39,33,45,44,48);(2,8)(3,12)(4,5)(9,10)(11,15)(13,14)(17,20)(18,25)(19,29)(21,36)(22,37)(23,35)(24,39)(26,44)(27,33)(28,45)(30,47)(31,34)(32,46)(38,43)(40,42)(41,48);(1,4,6,5)(2,8,10,9)(3,13,14,12)(7,15,16,11)(17,21,22,23)(18,26,27,38)(19,30,32,31)(20,35,37,36)(24,40,41,28)(25,43,33,44)(29,34,46,47)(39,45,48,42)

Polycyclic presentation:

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

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, 151101);