#243917

order	512
nilpotency class	3
isoclinism	isoclinic class of #232771
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

681 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=ef,c^2=h,d^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,d^c=dh,e^c=ei,f^c=fi,g^c=gh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151113);