#339819

order	512
nilpotency class	3
isoclinism	isoclinic class of #339775
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	#3151	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

590 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,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=dei,e^b=eh,g^b=gh,f^c=fi,g^c=gh,f^d=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 247015);