#333172

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

601 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)(3,5)(4,12)(6,8)(7,10)(9,15)(11,14)(13,16)(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);(1,3,5,2)(4,9,13,11)(6,8,7,10)(12,14,16,15)(18,24)(19,30)(20,34)(21,23)(26,42)(27,28)(32,37)(35,36)(38,41)(40,45)(43,48)(44,46);(2,8)(3,10)(6,7)(9,16)(11,12)(14,15)(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);(1,4)(2,9)(3,11)(5,13)(6,15)(7,14)(8,16)(10,12)(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,b^2=i,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, 240368);