#333150

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	#3253	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

589 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:

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

Polycyclic presentation:

2^9|a^2=ehi,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, 240346);