#6437331

order	512
nilpotency class	3
isoclinism	isoclinic class of #6433049
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	#2856	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1345 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)(11,14)(12,16)(13,15)(17,20)(18,21)(19,22)(23,24)(25,26,30,34)(27,35,38,32)(28,29,33,31)(36,39,40,37);(2,5)(6,8)(9,11)(10,14)(12,18)(13,17)(15,20)(16,21)(19,23)(22,24)(25,27,30,38)(26,32,34,35)(28,36,33,40)(29,37,31,39);(11,17)(14,20)(18,23)(21,24)(25,28,31,26,30,33,29,34)(27,36,39,32,38,40,37,35);(1,3,4,7)(2,6,5,8)(10,15)(14,20)(16,22)(21,24)(25,29,30,31)(26,33,34,28)(27,37,38,39)(32,40,35,36);(9,12,13,19)(10,16,15,22)(11,18,17,23)(14,21,20,24)(27,38)(32,35)(36,40)(37,39)

Polycyclic presentation:

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

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, 6344527);