#6436316

order	512
nilpotency class	3
isoclinism	isoclinic class of #6432861
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1365 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)(27,33)(28,35)(29,39)(30,36)(31,40)(32,37)(34,38);(2,5)(6,8)(9,11)(10,14)(12,18)(13,17)(15,22)(16,21)(19,24)(20,23)(25,27)(26,33)(28,31)(29,30)(32,38)(34,37)(35,40)(36,39);(11,17)(14,21)(18,23)(22,24)(25,28,32,29)(26,34,37,33)(27,30,38,31)(35,36,39,40);(9,12,13,20)(10,15,16,19)(11,18,17,23)(14,22,21,24)(25,29,32,28)(26,35,37,39)(27,30,38,31)(33,40,34,36);(1,3,4,7)(2,6,5,8)(25,30,32,31)(26,36,37,40)(27,28,38,29)(33,35,34,39)

Polycyclic presentation:

2^9|a^2=hi,c^2=i,d^2=h,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,c^b=ch,e^b=ei,g^c=gi,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, 6343512);