#243910

order	512
nilpotency class	3
isoclinism	isoclinic class of #232766
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

664 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=ef,c^2=h,d^2=h,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dh,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

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151106);