#243890

order	512
nilpotency class	3
isoclinism	isoclinic class of #10012
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#399
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

679 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:

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

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,e^c=ei,f^c=fi,g^c=gh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151086);