#243898

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	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

666 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:

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

Polycyclic presentation:

2^9|a^2=eh,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,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, 151094);