#6497398

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470874
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×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1478 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)(3,6)(4,5)(7,8)(9,10,13,16)(11,14,17,21)(12,19,20,15)(18,24,23,22)(25,26,30,35)(27,36,38,32)(28,31,39,40)(29,33,37,34);(2,5)(6,8)(9,11)(10,14)(12,18)(13,17)(15,22)(16,21)(19,24)(20,23)(25,27,37,39,30,38,29,28)(26,31,33,32,35,40,34,36);(1,3,4,7)(2,6,5,8)(11,17)(14,21)(18,23)(22,24)(25,28,29,38,30,39,37,27)(26,32,34,31,35,36,33,40);(11,17)(14,21)(18,23)(22,24)(25,29,30,37)(26,33,35,34)(27,28,38,39)(31,32,40,36);(9,12,13,20)(10,15,16,19)(11,18,17,23)(14,22,21,24)(27,38)(28,39)(31,40)(32,36)

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=f,d^2=i,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^b=ei,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, 6404594);