#6499482

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470374
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#34624	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1531 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fg,c^2=gh,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,d^c=di,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6406678);