#339761

order	512
nilpotency class	3
isoclinism	isoclinic class of #339703
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2675
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

557 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)(3,10,12,16)(4,7,14,8)(6,13,15,11)(17,18)(19,27)(20,33)(21,25)(22,37)(23,29)(24,28)(26,36)(30,41)(31,45)(32,40)(34,48)(35,43)(38,44)(39,47)(42,46);(17,19,23,31)(18,25,29,40)(20,34,36,47)(21,22,32,24)(26,41,33,42)(27,28,45,37)(30,38,46,35)(39,43,48,44);(1,3,5,12)(2,6,9,15)(4,13,14,11)(7,10,8,16)(17,20)(18,26)(19,30)(21,34)(22,35)(23,36)(24,38)(25,39)(27,42)(28,43)(29,33)(31,46)(32,47)(37,44)(40,48)(41,45);(2,7)(4,14)(6,10)(8,9)(11,13)(15,16)(17,21,22,31,23,32,24,19)(18,27,28,40,29,45,37,25)(20,34,35,46,36,47,38,30)(26,42,43,48,33,41,44,39)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 246957);