#333168

order	512
nilpotency class	3
isoclinism	isoclinic class of #333127
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2669
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3548	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

593 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=hi,c^2=h,d^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,e^b=eh,g^b=ghi,f^c=fi,g^c=ghi,e^d=eh,f^d=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 240364);