#243896

order	512
nilpotency class	3
isoclinism	isoclinic class of #232760
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	#123
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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151092);