#333161

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:

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

Polycyclic presentation:

2^9|a^2=e,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, 240357);