#6386390

order	512
nilpotency class	3
isoclinism	isoclinic class of #6342800
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×C₂²⋊C₄
commutator	C₂³	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3518	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2931 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,e^b=ei,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6293586);