#175487

order	512
nilpotency class	3
isoclinism	isoclinic class of #174829
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

706 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,d^2=hi,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=di,e^a=eg,f^a=fi,d^b=dg,e^b=eg,f^b=fh,e^c=eh,f^c=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, 82683);