#6752368

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682564
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1330 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=i,c^2=fg,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=di,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6659564);