#6499486

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	#35278	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1538 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)(27,29)(32,34)(33,38)(35,37)(39,40);(2,5)(6,7)(9,11,13,17)(10,14,15,21)(12,18,20,23)(16,22,19,24)(25,26,27,32,30,34,29,31)(28,33,35,39,36,40,37,38);(1,3,4,8)(2,6,5,7)(10,15)(11,17)(16,19)(18,23)(25,27,30,29)(26,32,34,31)(28,35,36,37)(33,39,40,38);(11,17)(14,21)(18,23)(22,24)(25,28)(26,33)(27,35)(29,37)(30,36)(31,38)(32,39)(34,40);(9,12,13,20)(10,16,15,19)(11,18,17,23)(14,22,21,24)(28,36)(33,40)(35,37)(38,39)

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,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, 6406682);