#6436512

order	512
nilpotency class	3
isoclinism	isoclinic class of #6433057
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

1338 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)(26,31)(27,35)(30,33)(32,36)(37,39)(38,40);(2,5)(6,8)(9,11)(10,14)(12,18)(13,17)(15,22)(16,21)(19,24)(20,23)(25,26)(27,32)(28,33)(29,31)(30,34)(35,40)(36,39)(37,38);(11,17)(14,21)(18,23)(22,24)(25,27,33,39,29,37,30,35)(26,32,28,36,31,38,34,40);(9,12,13,20)(10,15,16,19)(11,18,17,23)(14,22,21,24)(25,28,29,34)(26,33,31,30)(27,36,37,40)(32,39,38,35);(1,3,4,7)(2,6,5,8)(26,31)(28,34)(32,38)(36,40)

Polycyclic presentation:

2^9|a^2=h,c^2=gi,d^2=hi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,c^b=ch,e^b=ei,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, 6343708);