#6499487

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470428
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	C₂×#540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1398 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)(26,31)(28,29)(32,34)(33,39)(35,40)(36,38);(2,5)(6,7)(9,11,13,17)(10,14,15,21)(12,18,20,23)(16,22,19,24)(25,26,29,31,30,32,28,34)(27,35,36,40,37,33,38,39);(1,3,4,8)(2,6,5,7)(10,15)(11,17)(16,19)(18,23)(26,32)(31,34)(33,35)(39,40);(11,17)(14,21)(18,23)(22,24)(25,27)(26,33)(28,38)(29,36)(30,37)(31,39)(32,35)(34,40);(9,12,13,20)(10,16,15,19)(11,18,17,23)(14,22,21,24)(25,28,30,29)(26,34,32,31)(27,36,37,38)(33,39,35,40)

Polycyclic presentation:

2^9|a^2=g,b^2=fg,c^2=ghi,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,e^d=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, 6406683);